Captain reported a loss of situational awareness when maneuvering at the minimum vectoring altitude while trying to locate GJT airport at night in mountainous terrain. The Captain expressed concern that this approach may not be safe.

Narrative

As we were nearing GJT airport; we requested the RNAV (GPS) Rwy 29 Approach. We were told by the ATC Controller that he would be unable to clear us for any approach to RWY 29 from our direction because their minimum vectoring altitudes were well above the charted approach altitudes. We queried him; because I did not believe that we would be unable to shoot either of the published approaches to 29; into an airport surrounded by high terrain at night. The Controller descended us incrementally as he could; but once we reached 9500 ft MSL; he stated that until we reported the runway in sight and cleared us for the visual; he was unable to clear us to a lower altitude due to terrain. As a result; we flew past the extended centerline of the runway and finally saw the airport as we passed approx. 5000 ft above the field elevation. I did not recognize the airport at first; because from our altitude it looked tiny; and I was looking farther to the north for the visual picture I expect when looking for an airport. Once we passed to the west of the airport and reported GJT in sight; the controller was able to give us vectors and a lower altitude. We were given a 180 degree turn for the downwind leg and then cleared for the visual approach; with turns onto base and final legs at our discretion.Given the limited moonlight; we could not see any of the terrain to the south or west of the field; only the outline of the mountain peaks against the skyline. I have approx. 12;500 hours of flight time; 10;000 with the airline and was extremely uncomfortable throughout this approach. Although I believe I was flying conservatively; I had no actual idea where the terrain was and just tried to fly the closest-in approach I could; while still getting configured in time to be stabilized for landing. To add to my mental workload; I was conducting IOE with a new-hire. I was mentally prepared to conduct the terrain escape maneuver if needed. One of the only things that gave me any comfort in conducting the visual approach is my experience level; conservative energy management and belief that ATC was watching us on radar and would have alerted us if they had gotten a 'low altitude' alert.I believe this operation is inherently unsafe: descending into a valley; surrounded by high terrain and just guessing what altitudes were appropriate while on a visual approach. I half considered diverting; thinking to myself; How is it possible that this is legal?" The airline should commission Jeppesen to create a custom RNAV STAR linked to an Approach for Rwy 29 that meets the criteria required by ATC."