Tower Controller reported a single engine aircraft landed with landing gear failure following two other incidents. Controller reported facility non compliance resulted in fatigue and working beyond legal maximum hours.

Narrative

I was working local at XA15 Z when Aircraft X on the visual approach said they had a gear issue and needed to go missed approach. I offered the aircraft a low approach offset to the left of Runway XXL so I could check the gear position. The Pilot flew toward the Tower and we advised that while the left Main and nose gear appeared to be down; the right main appeared to be up. The Pilot said that matched his indications. He stated he want to try to do an emergency extension of the gear. I asked if he wanted to stay with Tower VFR or go back to ZZZ1. He said he wanted to stay with me ; I told him to squawk VFR and coordinated with approach. I assured the Pilot I was going to sterilize the airspace to give him whatever he needed to work the issue. I offered to let him work the issue at 2000 feet above the airport to keep him over all of the Runways; winds were light. Aircraft X said there was 1 Soul on board and 4 hours of fuel remaining. This was relayed to the supervisor who rang the Primary Crash Alert Circuit. I gave the aircraft instructions to avoid traffic (I was still getting everyone out of the way). The Pilot troubleshot for a few minutes and then said he was ready to proceed inbound. The Pilot restated nothing worked to get the right main down. I offered one more flyby for us to check it in case the gear had come down during manual emergency gear extension attempt. He denied the request and was ready to start inbound. I asked if he planned to have all three gear up. He stated only the right main would be up. The Pilot asked if ARFF equipment was in place yet and I stated they needed another minute or two. The Pilot made a 360 and once the Fire Chief was on scene the Pilot asked to talk to him. I had ground push ARFF to Tower frequency. I gave ARFF the frequency and the Pilot of Aircraft X intimated that his big concern was 25 gallons of trapped fuel in the right wing creating a risk. He requested that ARFF units spray the right wing as needed. The aircraft touched down at XA28z; I authorized rescue personnel to chase the aircraft down Runway XXL; The aircraft dropped on to the right wing after a long rollout and turned about 90 degrees right; the ARFF units sprayed the wing immediately and we suspended ops. The Pilot stated he had shut everything down and got out. It was really amazing airmanship. As soon airport authority allowed we moved aircraft to other runways and resumed operations. Our staffing meant that when this incident I had had 2 minutes off between XY40z and XA30z. Prior to this incident we had two previous incidents that required reports. We are working 6 day work weeks often with longer than 8 hour shifts. While in this incident I was able to fight through the fatigue that can't always be the case. I had to immediately go back on position in local to keep a coworker from going over two hours. During the first part of this session I had an aircraft lose control and exit a runway into the grass. There was nobody available to get me out even if I wanted to get out. Then; this [incident] happened; I was already very tired. This trend at our field is going to get someone hurt. We cant run 2 hour sessions 6 days a week indefinitely without it having a detrimental impact on safety. So today I got to file 3 reports in the first 4 hours of a shift. We are exhausted.