It depends on which capacity you are speaking of and what the altitude of your target event is. If you are training for shorter high intensity events like 5-10,000m running race held at sea level then the majority of your training needs to be done a low elevations so you can train with sufficient intensity to stimulate the correct adaptations for such speeds. A stint of training at high elevation can help with sea level performance. The system of train low/sleep high has been used very successfully in many endurance sports for decades. In that system the sleep/live elevation is normally never more than about 8500ft or 2600m. Above that and sleep is not as restful and so recovery is more impacted negatively.
For mountaineering it is a bit different. I think you are conflating training adaptations (fitness gains) with altitude adaptations (acclimatization gains). Both will improve performance at altitude but they demand different stimuli that are to some extent counter productive to each other.
Fitness can be increased easier (better and faster) by training at low elevation. That’s because you can train at a higher intensity when more oxygen is available. You will also recover faster at low elevation so you can maintain a higher training load which in turn gives a bigger training stimulus.
Acclimatization on the other hand is best accomplished with prolonged exposure to high elevations.
While there will be some acclimatization effect from short exposure to high altitude during training, these will be insignificant unless frequent and prolonged. For example living at 6000ft and going to do a 3 hour hike at 10,000 feet 1x/week is not going to do much for your acclimatization and is definitely going to be less effective training that doing the same hike at 6000ft (same terrain profile).
Bear in mind that at high elevations like you are planning on acclimatization will play a huge role in your performance on those climbs. This is why supplemental oxygen is so popular on some 8000m peaks. It will allow a much less fit person to still be successful. There is no substitute (except for more supplemental oxygen) for proper acclimatization.