Air taxi pilot reported an unstable approach resulted in loss of control during landing roll and two flat tires. Aircraft stopped on runway.

Narrative

Received a pop-up trip to ZZZ; 2 days before departure. This 5;000' runway is situated in a valley; surrounded by mountains. My prep for flying to this destination (for the first time ever) was to use Foreflight's 3D mode to assess the environment around this airport. I was impressed at how deeply this runway was situated between surrounding mountainous terrain. This prompted me to call the Chief Flight Instructor at the destination airport/FBO to get an experienced briefing on how best to get in and out of this airport.Unfortunately; this Instructor had a cavalier approach to describing how to negotiate approaches to his airport. He started by saying that getting in was no big deal; just come in straight into Runway XX; just watch for the specific marker just before the approach end of Runway XX. He also admitted he was somewhat partial since he had been at this airport for a long time.This Chief Flight Instructor then added a story which did not provide any reassurance whatsoever for this upcoming flight. He described a landing at his airport prior by an aircraft which bounced multiple times; extremely hard; on Runway XX; bending the nose gear backward 45 degrees; damaging one of the main landing gear legs; blew out a main tire; flamed out the engine; and partially departed the runway to the right. He then added that the recovery aircraft had to do a Go Around on its first approach to Runway XX; and laughed as though it was a 'fail' that they went around.The next interesting occurrence was that my lead passenger on this Part 135 flight to ZZZ from ZZZ1 decided to sit up front with me and assured me that he had flown into this airport a lot in his own aircraft and other planes with no problems; even with a steep approach to Runway XX. When I mentioned the potential for a 'Go Around;' he was visibly surprised as he asked why. I explained to him that a 'Go Around' is free; and is sometimes necessary for our safety. I also thanked him for sitting up front with me; because it would be helpful to have another set of eyes to identify the airport as soon as possible. To this he replied; 'why; it's gonna be right there.' So I dropped the subject as I realized he didn't have an appreciation for any layman version of CRM.ATC kept us at XXXX' to clear us over the last mountain ridge before entering the airport environment; lining us up perfectly for a visual final approach to Runway XX. After clearing the ridge; we were cleared to XXXX'; which is when I spotted the runway as I began the descent. I informed ATC that we had ZZZ in sight. ATC released me to local comms and I advised ATC that I would cancel IFR on the ground as I felt that this was the most prudent way to cancel (I've also routinely canceled on the ground at my home airport; ZZZ1; to add an additional layer of safety). At this point; I disconnected the autopilot; thrust to idle and maintained level altitude long enough to lower landing gear and add 50% flaps. Interestingly enough; the auto-throttles had not automatically slowed the airplane yet; as close as it seemed we were to the airport. That may have been due to altitude. The descent to the approach end of Runway XX was steep; but that was to be expected from the 3D analysis and the Chief Flight Instructor.The problem was that once we leveled off and added full flaps just before negotiating the 'specific marker' at the approach end of Runway XX; we were still faster than normal; but slowing. As we touched down smoothly just before the touchdown zone; I realized that my rollout speed 'felt' significantly faster than any other rollout I had ever had in this aircraft. Smooth brake pressure was applied with no appreciable slowing noted. I tried pumping the brakes and; as the end of the runway was in sight; I changed to full and constant heavy brake pressure as I was now committed to stopping before running off the end onto a descending field and into a hazardous environment. The rollout was initially on the center line. As we slowed to a stop with a little less than 1000' remaining; directional control became more challenging as the plane swerved left and right to a stop. It was at this point that I realized that 'I had been here before;' as in 'simulator demonstrations of blown out tires.' The difference was that in the Sim; I was not at the controls; so I did not have a feel for it; except for the swerving at the end. We stopped just to the left of the center line. When I added some thrust; the airplane didn't move as expected; so I did not force the issue. I called the FBO and announced my concern for 2 flat tires and that I would need assistance for aircraft movement and getting my passengers off the plane and off the field.The lineman at the FBO had already closed the runway. In the meantime; I called ATC to close my IFR flight plan; and they confirmed that they received a call for the runway closure.The lineman brought the passenger's transportation plane side; the lead passenger and I exchanged phone numbers; discussed several scenarios for them getting back home; and I then immediately called our Director of Maintenance. He had the opportunity to speak with the FBO's A&P.I then discussed the plan per the lineman and the FBO's A&P for safely moving the plane off the runway. Our Company's concern was to prevent any further damage to our plane as there was no apparent damage noted except for the tires. I even offered to run into town to purchase some heavy-duty vehicle tire dollies for us to use but; in the end; the A&P safely removed the plane onto a taxiway without causing any damage to the wheels by slowly and carefully using a mule tug.With excellent teamwork and coordination by our Director of Maintenance; another aircraft was flown in from my home airport by one of our best pilots with a phenomenal maintenance tech on board to replace the tires and allow me and my passengers to return home safely in another aircraft.The lessons learned from this event are quite clear. There is absolutely no room for continuing an approach to a landing that is anything other than stable in every way possible. In this case; unfamiliarity with the field and surrounding environment was a contributing factor. You would think that a thorough consultation with a reputable pilot from the destination field would have been beneficial but; in this case; it fell short.Identifying the runway late and at high altitude contributed to a worsening of the high descent rate (expected due to the specific marker before the runway; but made worse by the late identification of the runway).The higher than normal speed at the threshold is another very important indicator for a Go Around at that moment. The hope was that there would be more dissipation of speed during the flare; but that didn't seem to happen to any significance. Any tailwind in this case would have made everything worse; but the winds were calm.Even after touching down; and especially at the perceived high speed; a touch and go was still another option. But there was a bit of concern; in the split-second instant; that it might not be a good option with whatever runway was left.The final option was what actually happened which; in retrospect; should not have been an option at all; any of the aforementioned other options were all better choices. Well; technically; only the Go Around options were acceptable. Other good options would've been S turns; circling over the airport to lose altitude; and circling south east of the airport where there was a break in the mountainous terrain.Although the mountainous surrounding terrain was somewhat intimidating; it shouldn't have been a concern because I could've simply climbed runway heading to whatever safe altitude I needed. Note: the MSA for the entire area centered on the MAP for the RNAV (GPS)-A approach is XXXX'; airport elevation XXXX'; In addition; I had tankered fuel from the departure airport; so I had plenty of fuel to flyplenty of circles.In the end; I am grateful that; at the very least; center line landings and center line control was emphasized in my training. That certainly contributed positively to minimizing damage to only rubber tires; and no metal damage. I am thankful for that training; it certainly produced results in this case.This experience was also a solid reminder of the need to Always; always; always have a low threshold for Go Arounds when something is not right with an approach. It's always been my mantra; but somehow got lost in the events described above.With all these issues experienced at this airport; ZZZ; over the years; I wonder if our company might consider this airport as a 'Special Use Airport;' requiring a briefing/training with a Chief Pilot or Check Airman; and maybe even a prior trip to this airport with an instructor; or with another pilot with experience landing there; before being cleared to fly into ZZZ solo or with a pilot who has never flown in there before.