Cessna 172 pilot reported a loss of aircraft control during landing and let the aircraft roll past the runway edge into the grass.

Narrative

While conducting a daytime VFR flight into ZZZ airport for the purpose of Commercial rating time building; I experienced a near-loss of control during landing. Rwy XX was in use for several departures ahead of my arrival; and I was instructed by Tower to expect straight in for Rwy XY to land. The winds were reported 040 @ 06kts; with no gusts reported. This introduced a light; but present; crosswind factor to the landing. The particular Cessna 172 aircraft I was flying for the day had a rudder trim control installed; and the aircraft had a fairly pronounced natural right-yaw tendency; enough to remove the ball from the center marks of the inclinometer almost entirely even during straight and level flight. As part of my landing brief/checklist; I noted this and elected to recenter the yaw trim; so as not to have any input on the control surfaces in a crosswind that I did not directly have control over. On short final; I experienced several gusts of near-direct crosswind; but nothing so severe as to merit the need for a go around; as they were very brief and relatively mild. At just moments; what must have been only a couple of feet away from touchdown; another gust of crosswind struck; the airplane weathervaned and also winged over to the right; and the right main gear made premature contact with the runway surface. This caused an exaggerated right-yaw off of axis as the rest of the aircraft settled onto the surface; and I believe the aircrafts pre-existing tendency of right-yawing rudder may have exaggerated the unexpected circumstance even further. I made smooth and steady left rudder input to correct; and as I did; the airplanes nose swung around to the left with far more intensity than I could have anticipated. I was careful not to 'stomp' rudder; and though I could have potentially inadvertently applied some left brake; I am nearly certain that I hadn't; as I am always mindful of inadvertent braking during landing. At any rate; the exaggerated oscillating/yawing motion to the left put the airplane into a skid down the runway at approx. 40-50 degrees off of center axis with the runway. As soon as I identified the point at which the plane was at risk of departing a controlled state; I insured that power was completely to idle; and my feet were well clear of any kind of brake input; as I felt that either at that point could exaggerate a loss of control or even cause the airplane to roll or flip. I used rudder to regain directional control as the aircraft lost energy; and back pressure on the yoke to induce aerodynamic braking and hasten the loss of momentum. At the point that I had regained safe and positive control of the aircraft; I was at the runways edge; and in order to continue to maintain safe control; I let the aircraft continue rolling straight ahead into the grass; at which point I introduced brakes to bring the airplane to a stop. Immediately; I made contact with ZZZ Tower to advise them of my status; and taxied straight ahead onto the taxiway ahead of me; at which point I stopped the aircraft entirely. Airport Rescue & Firefighting (ARFF) was dispatched as a precaution. Upon the arrival of ARFF; I was able to make contact with them via Ground frequency; advise them I had no injury onboard; and as far as I could tell at that point I had not experienced any kind of prop strike; or that any part of the aircraft had made any kind of damaging contact with the surface or any elements of the runway lighting etc. They advised no sign of damage from outside; and I was able to taxi to park under my own power. Upon parking and shutdown/securing of the plane; I performed a thorough inspection of the aircraft along with ARFF; and located no damage; or sign of damage. At no point did I experience any hard strikes to any component; simply a near-loss of controllability. At this point I made immediate contact with ZZZ Airport Management; and also contacted ZZZ Tower as a courtesy; providing my certification; contact info;and statement on the incident to all parties. Airport Management advised there had been an inspection of the landing runway and no damage to any elements of the airport or runway; and no sign of any kind of prop or aircraft strike was found; and they did not expect that any formal investigation etc would be necessary. I made contact with the owning/operating flight school and advised them of the situation; and they advised that if I was safe; the aircraft was safe and undamaged; and if I was comfortable with making a return flight; that I should do so. Upon refueling the plane; I once more conducted a thorough preflight inspection; and satisfied that I was safe; and the aircraft was safe; I moved on to startup; and conducted a thorough runup. There was no sign of damage or malfunction in any regard; and again; I had not experienced any instance of making any undue or damaging contact with the surface or runway elements at any point during the incident. I elected to continue with the return flight to the home airport of ZZZ1. I made a short delay on the departing runway to make a final runup; and commenced the return flight. The return flight and landing were made safely and without incident. Upon making contact with the school; the aircraft was taken off the line for the soonest possible inspection to be made; and I also advised of the situation with the abnormal rudder control. One of the instructors added that he had been in the same plane with a student the day prior and had also noted the planes hard nose-right tendency while in center trim. In hindsight; I believe that the planes tendency to 'weathervane' moreso than is typical may have been a contributing factor to the exaggerated effects I experienced during the incident. The school advised the concern regarding the rudder would be made known to maintenance during the inspection to address any potential safety concerns surrounding it; and prevent any future potential incidents.