George, Roughly half of the mountaineering athletes I work with are runners and half are not. Both groups train effectively with some consideration. For me the best thing about running is the simple ability to train at a target HR in almost any terrain if need be. It is almost functionally impossible for a very fit individual to keep their HR in Upper Zone 2 for an extended period of time on flat ground with a walking gait. A runner can do that easily. Thus the non-runner needs to find hilly terrain and/or use gym based machines to hit target HR. As you mention above running actually isn’t perfectly sport specific for carrying a big pack at high altitude so it can’t be the totality of training in the later phases. Even the most dedicated runner will want to transition a good sized chunk of their training volume to hiking centric training in order to be the best prepared. A lot of runners will mix run/hike outings on rolling terrain where they’re hiking the ups and jogging the downs and flats to keep their HR in the desired zones with other days where they are hiking with a weighted pack for sport specificity. This mix will produce solid results for most folks who like to run as part of their program.

George, For my athletes that enjoy running I like program the weekend such that one day (usually Sat) is a running day (may involve hiking the uphills) and is, of course, unweighted then Sunday is a hiking mode day usually with some pack weight. As an aside I do think in particular train running can have some benefits with movement pattern and agility. As we age if we just walk slowly on very groomed terrain (I.e. Treadmill etc) we really lose the ability to move nimbly and efficiently on uneven terrain. A bit of uptempo movement like trail running seems to combat this and keep athletes sharper.