C172 Flight Instructor reported experiencing a rough running engine while in the traffic pattern. Instructor returned and blew a main gear tire during landing.

Narrative

I have no knowledge of any aircraft damage occurring during my flight. To the best of my recollection; I am providing this statement regarding a failure to properly preflight an aircraft. While landing at ZZZ airport; I experienced a mechanical failure which led to a loss of control of the aircraft during touchdown. I was able to regain control of the airplane without colliding into any other objects; persons; or aircraft. During touchdown; the main wheels touched down first; but as soon as the nose wheel came down; the plane veered sharply to the left. Though throttles were already at a low RPM; I further reduced thrust and applied full right rudder to correct the track towards the runway centerline. With very little rudder authority; the plane continued to the left. I applied differential power on the left engine to counteract the left turning force of the aircraft; and correct to the centerline. At no time; by my estimation; did any of the aircraft wheels leave the ground during this correction maneuver. Once the aircraft was under control; I taxied the plane to the ramp but the right rudder was unresponsive to normal inputs. During the landing and throughout the flight; the Owner of the aircraft and Flight School; provided flight instruction from the back seat of the aircraft and acted as a CFI. The Owner was always fully engaged as an instructor with both my flying partner and me; particularly during the take-off and landing phases of flight.After securing the aircraft the Owner accused me of a prop strike. While I could see the right propeller was bent on one tip; there is no supporting evidence that a prop strike occurred during my flight. BACKGROUNDTaxi and Ground Operations: Before taking off at ZZZ1 on Day 3; I experienced notable difficulties taxiing the aircraft; the controls felt abnormal and 'mushy;' however; I excused the difficulties as unfamiliarity with the aircraft Aircraft X. It was my first time flying that plane. On Day 0; I completed a checkout flight with the Owner and a time building flight later that day with my flying partner in the right seat of a different PA30 (Aircraft Y). Upon my arrival to the hangar on Day 3; I was surprised to learn that Aircraft Y was not the aircraft that I would be flying. The Owner told me that Aircraft Y was actually not a part of the flight school's fleet. I was then presented with Aircraft X and told by the Owner that I did not need to conduct a preflight of Aircraft X because the preflight had already occurred and the Owner stated that the fuel was full. The Owner said that we could just 'hop in and go.' I still decided to complete a brief walk-around; and my flying partner did the same upon their arrival to the airport; shortly after me. During the taxi at ZZZ1; the Owner noted to my flying partner and me that I would need help with directional control and that he would need to assist on the rudders on Aircraft X. It is important to note that Aircraft Y had been in maintenance prior to my flight on Day 3. The Owner informed me that a test flight following maintenance on Day 2 had been conducted. However; there is no evidence that a post-maintenance flight occurred. The alleged flight does not appear on logs. In fact; the last recorded date for Aircraft Y was a month prior. I transcribed all ATC communication archives at ZZZ1. I found no record of anyone calling for Aircraft X taxi clearance at ZZZ1 the day before. I focused on this time period because the Owner stated both in person and via text message that they personally flew Aircraft Y after its scheduled maintenance the day before. Cruise Flight: During cruise flight on Day 3 I pointed out to my flying partner several times that Aircraft X was exhibiting negative static stability. Without a constant application of upward force on the left wing; the plane banked drastically to the left. Later in the flight; as I voiced this concern again; the Owner finally pointed out that the plane had arudder trim. I felt relieved and fathomed the aircraft would be easier to control on the ground now that it was properly trimmed. I did mention I was unsure how to set the rudder trim to neutral; but the Owner never explained what a neutral setting would look like; until after the flight. The Owner seemed unsure about how to properly use the rudder trim and it was not until we were on the ground that the Owner explained how to know if the rudder trim was set to neutral. The Owner appeared to have learned that information after speaking with maintenance. The head of maintenance at the Flight School also happens to be the Owner's spouse. The Owner further stated the overall assessment that I only used a 'little bit' of rudder trim during the flight at cruise and only 'took a little bit out' prior to landing. Landing and Post-Flight: I deviated from the center line promptly after touchdown due to a mechanical failure; but I effectively recovered control of the aircraft without any damage occurring to the aircraft. As we taxied to the fuel station; the Owner first directed us towards the fuel station; so we could refuel for the next flight. This is important to note because refueling for the next flight; bears the assumption that the plane was still safe to fly. The plane did not bounce after touchdown and no one onboard the aircraft heard the loud sound of metal against the pavement. After we shut down the engines; as we refueled; I expressed my unwillingness to fly the plane again until maintenance could address my concerns. I also made the suggestion to the Owner Aircraft X would need to be taken for a test flight to see if the issues I experienced could be replicated. The Owner; through their actions; refused to restart the plane or acknowledge the concerns I voiced. The Owner; instead; began inspecting the right propeller and stated we needed to await a reply from maintenance regarding the airworthiness of the plane. I was perplexed as to why the Owner was focused on the propeller; I had not yet seen the damage at that time. The Owner disregarded the reported loss of control at touchdown. My flying partner said quietly to me; the Owner was 'focused on the wrong thing.' Instead of investigating the issues raised concerning the inability to effectively taxi and lack of right rudder effectiveness; the Owner was focused on the right engine and miraculous bend on the right propeller. After securing the plane; my flying partner and I agreed we would not fly the aircraft again since it was apparent to us the Flight School planned to replace the propeller and would not give any thought to my concerns regarding what could have caused the plane to veer to the left after touchdown. My flying partner and I sat in the FBO at ZZZ for at least 20 minutes and waited for the Owner to come inside. The Owner sat outside on the pavement; underneath the wing. Eventually; I booked a rideshare at ZZZ to take us back to ZZZ1. During the ride home; the Owner; from the front seat of the rideshare; turned back to look at me and stated; 'my [spouse] says you need to file an insurance claim.' I explained to the Owner that I would not file a claim until maintenance determined what could have caused the plane to veer so violently off of the centerline. If a propeller strike had occurred; I postured; how did we not hear the sound of a rotating metal blade hitting the pavement or perceive that a prop strike had occurred? The Owner offered several nonsensical theories: (a) that I may have been on the brakes too soon; (b) that the propeller hit a runway light; and (c) the propeller hit the pavement during the recovery (I was able to disprove these 3 explanations the next day). The Owner could sense that I was upset and assured me everything would be okay; saying; 'it happens all the time; it's happened to me beforeGǪ it's a good thing you have insurance.' I told the Owner that I felt set-up and stated very clearly; I would not file the claim. Itold her it was unfair that I was the first person to fly the plane after it had been down for maintenance. The Owner rebutted that Aircraft X did have a maintenance test flight the day before and the Owner had personally determined everything was fine. I do not believe this flight actually happened because I have found no evidence of it. ADS-B is required to operate in the Bravo airspace at ZZZ1 and all ATC communications are recorded and broadcast publicly at ZZZ1.According to the airport manager; the flight instructors and other witnesses at ZZZ who saw my landing felt that the approach was stabilized and that it appeared to be a loss of rudder control when the plane touched down. I heard no report of heavy or unusual braking upon landing or of anyone hearing a propeller strike. The airport manager and I also inspected the runway. We did not find any braking skid marks leading left off the runway centerline that might indicate heavy braking upon landing. I wanted to know if any runway lights were damaged; but there are no lights on runway XX; except for the PAPI lights (which were fully operational). We also inspected the runway for scratches on the pavement to support the claim that the right propeller hit the runway after landing. We found no such evidence. MAINTENANCE DELAYS / EXTERNAL PRESSURES On Day 1; the Owner said the Aircraft X needed an oil change and on Day 2; the Owner said the right brake was leaking fluid. Both issues took a full day to resolve and we did not fly at all for those 2 days; nor did I know at the time the Owner was referring to maintenance for Aircraft X; a plane I had not seen before. I extended my rental car and hotel reservations an extra day due to these maintenance delays; in hopes of getting the remaining flight hours as agreed upon with the Flight School. Both my flying partner and I had traveled from out of town to build multi engine time. I was disappointed to not fly for 2 full days of my planned time building trip. On the day of the improper pre-flight; there was a lot of external pressure to fly. I believe these external pressures to fly impacted my pre-flight procedures. CONCLUSION: My investigation into the alleged prop strike leads me to conclude the damage on the right propeller tip was pre-existing. I also noticed that Aircraft X flew almost every day. There is no ADS-B record of the plane flying again until the event; the day of the incident in question. This to me; demonstrates negligence on the part of the flight school. I should not have been the first person to fly Aircraft X after such an extended period of the plane being grounded and undergoing maintenance. Furthermore; if the plane was flown at any time not depicted online; it would mean the transponder was not turned on at ZZZ1 within the Mode C veil; which is a blatant violation of FAR/AIM regulations.