A TRACON Controller reported a Corporate departing an airport where the Tower closed early due to staffing turned the wrong direction and flew below the minimum vectoring altitude and into confliction with VFR overflight traffic.

Narrative

I was working the ZZZ sector. ZZZ Tower had gone home early due to staffing on this day and declared ATC zero 3 hours earlier than their normal closing time. The Flight Data Controller issued a clearance to Aircraft X; for the ZZZZZ departure of of ZZZ; climbing to 6000 ft. and the aircraft stated they were departing Runway XX. I had some traffic in the area of ZZZ; so when the Supervisor requested release for the aircraft; I released them and ensured my traffic was out of the way of the left hand climbing turn on the ZZZZZ departure off of XX. I had two aircraft South of ZZZ that I vectored east of the field and then gave own navigation; and climbed another VFR to at or above 055 holding over the ZZZ VOR in a published hold to alleviate any potential conflict of the initial climb before the left turn required of the Aircraft X at 500 ft. to join a radial outbound from ZZZ; which would take the aircraft South of the VOR to join.Aircraft X checked on moments later somewhere out of 2000 ft. in a 3200 ft. MVA; and I observed the aircraft making a right turn; not in compliance with the departure procedure. I immediately issued a low altitude alert to the aircraft. I had the two aircraft at 4500 ft. East of the traffic; with the further North one in conflict. I initially wanted to stop the Aircraft X at 4000 ft. to alleviate this conflict; but the aircraft appeared to have turned direct ZZZZZ1 and towards a 4700 ft. MVA so I did not think I would have enough time to do so and not enter that higher MVA. I immediately utilized my map and issued a heading of 170 for terrain to the aircraft; which also alleviated the conflict with the Aircraft Y and called the traffic to Aircraft X. The vector made the Aircraft X pass about a mile West of the Aircraft Y; and I observed it deconflicting with the next Aircraft Z 2 miles South as well.I called traffic to my other VFR maneuvering over the VOR at 5500 ft. as well; and they replied in sight; and I advised the Aircraft X to expedite climb to 6000 ft. Then decided to try to avert a TCAS RA as ZZZ penalizes these occurrences even when legally separated by outer class C requirements; and assigned 5000 ft. Aircraft X read it back after questioning if it was for them; but still went through the altitude. I continued to climb the aircraft to 9000 ft. and then 19000 ft. and issued the brasher to the aircraft once the situation was stabilized. I missed all VFR traffic with Aircraft X that I had intentionally positioned to the east side to de conflict with Aircraft X required to turn to the West that they did not execute. The pilot called within 5 minutes while still airborne and told the OM they had a TCAS alert and elected to turn right instead of left (which does not make sense as my understanding of TCAS is it provides vertical guidance only); and after reflecting on this statement overnight; I believe the pilot was referring to my turn to the 170 heading which should have been a right turn anyways; completely not realizing that they never complied with the departure procedure and instead went direct ZZZZZ1 immediately after departure.There are a lot of factors that played into this event; and a lot of things that could have resulted in a safer scenario. In my opinion; MOST IMPORTANTLY; the pilot should have executed the ZZZZZ departure as assigned. There would have been no conflicts at all; and no IFR aircraft below the MVA; had the pilot complied with their clearance. The following points are only what I could have potentially done better in recovery of an unsafe situation that should not have developed in the first place. By the time Aircraft X had checked on; I already had observed their turn to the right; the wrong direction for this departure. Unfortunately; the MVA's rise rapidly and there are mountains all along the east side of the ZZZ airport. I wanted to turn the aircraft to the left but already knew the time to roll out of the turn and into the left turn would take too long and could take them into higher terrain. I issued a low altitude alert; and observed their climb rate and direction of flight; and quickly realized they were not turning southeast down the valley for the radial off of ZZZ intended to keep them away from terrain; so I knew they weren't flying any part of the departure; not just turning the wrong direction. At that point I knew for the safety of flight I had to issue the vector southbound away from both terrain and traffic. I issued a traffic call but no traffic alert. I knew my turn would in fact keep the aircraft away from the Aircraft Y. Although per 7110.65 requirements I did not believe the aircraft would end up in an unsafe proximity because I had already resolved the situation with the heading and observed it; I understand that the people that review events in the region expect these any time a conflict alert goes off. I fully realize that this is to enhance safety and I will try to issue it in the future during an unexpected event. I did not call traffic for the Aircraft Y as I knew I had a lot to do with Aircraft X at the time because they were heading towards two additional aircraft after passing the first Aircraft Y. I should have tried to give a quick call to them as well.When I attempted to stop Aircraft X at 5000 ft. after issuing 6000 ft. I should have accepted in my recovery that these aircraft were also going to miss by legal separation and ZZZ obsession with TCAS RA's is not a reason for me to not execute my personal best judgment that getting Aircraft X higher and out of the situation sooner would be a better solution than trying to avoid a nuisance RA. The pilot was unable to comply with instructions anyways as I had changed them too many times and I am sure they were completely behind the aircraft due to the numerous issued instructions and traffic calls I had to give them to fix their situation. It would have been much better for me to stick to the original instruction of 060 or to just climb the aircraft to 090; as I did moments later. I believe that most of the time my recoveries are on point with fewer transmissions and traffic alerts even when alleviated most of the time. I did feel that this situation required me to engage quickly and my reaction time was slightly slower than normal; which is extremely disappointing to me as an air traffic professional who strives for as close to perfection as possible in my work. I cannot honestly determine what caused this; but the only possibility I can think of is the amount of time and work I had done that day and the week this occurred. I had been heldover on two hours of overtime for staffing as we were down 3 Controllers on the swing shift; and this was my last sector rotation of the day on the 10-hour shift. It had been a busy day from about XA:30AM onward on every rotation; and most of my rotations had taken me close to two hours on position because of staffing; and a push by ZZZ management to get Controller time-on-position as high as possible. I had a quick turn the night before to a day shift; and had started my workweek on a mid Sunday morning; overtime swing Monday; day off Tuesday; and then back on Wednesday. I had trained someone for over 5 hours on Thursday night as well before my quick turn.I believe it is always a requirement to perform at your highest level despite any obstacles and high workload that we encounter as Controllers; but I do believe it's possible my reaction time was just a split-second slower than normal due to the schedule and time-on-position push from management that has developed in the past year in addition to a lack of staffing and 6-day workweeks increasing the fatigue for me and our entire Controller workforce. I hope this is not the case; but there is only one explanation that I can give for this. Finally; the ZZZZZ departure needs to be addressed. Pilot deviations on this procedure are not an overly rare occurrence. I cannot tell if it is because of the amount of GA traffic; but it is also a very 'busy' diagram of the procedure. The narrative for each runway is clear; but to me it seems that the pilots are not reading the narrative or unconsciously ignoring it and proceeding on course anyways. In addition; because ZZZ Tower had closed early; there was no Tower Controller available to observe and fix the initial wrong turn by this aircraft. The Control Tower is a valuable resource and needs to be staffed appropriately; as do all facilities in the national airspace system to ensure the safety of the system. All of these factors are known; yet all of the factors remain unresolved. There needs to be more pressure to fix these issues before they become catastrophic events. I made every effort to fix a situation that developed because of the pilot deviation by Aircraft X to alleviate conflict with terrain and three other aircraft. I will continue to do this in the future to the very best of my ability and will constantly try to improve and incorporate a traffic alert into my instructions to further enhance the necessity of my instructions to unstable aircraft in the future.