For what it's worth, 8 years ago when I had a climbing coach and took it seriously we did HIIT (before it took off as a trend) and it was further super tailored for climbing - for instance had days of the week called climbing specific which were mostly campus board with camera work for motion tracking type stuff. So at least I can vouch for a small group of fairly serious Colorado climbers who mostly did this type of training versus just running for many miles. Outside of climbing / training I mostly just walked a LOT to stay lean but that's quite different than running for long periods. I think there is some specific differentiation between alpine sport climbers and boulders/indoor comp climbers in terms of endurance needs.

I definitely made noticeable strength gains with his program I did it from like 18-23/4. Biggest regret of my life giving it up but that's another story!