In our TftNA book we propose doing weight bearing training as the best method when training for alpine climbing or mountaineering.
First, let me address the notion of Specificity: foot borne activities like running, hiking, skiing (either skimo or XC) all require the postural muscles to be engaged and used in locomotion. Exercises like cycling or rowing, where you are seated do not have the same muscle mass requirements for resisting gravity’s force so require less work. Cycling is an amazingly efficient means of human powered locomotion but that efficiency is also why it is not the best training for anything other than cycling. Similarly; swimming, while engaging a large muscle mass relies on the water to support the body in a (normally) prone position. This means that more of the muscle work can go into propulsion and less into postural control fighting gravity. In swimming the heart has to work less hard to move the blood due to the horizontal body position and lack of gravity. Not only that the water dissipates the heat of muscle work better than air does allowing less of the blood to be need for cooling and more to be used for work.
So, training specifically for your sport will always give better results than non-specifically. This is especially true for more advanced athletes.
What you are faced with is a common complaint. How to do all the needed foot borne training without beating your body into a state of chronic injury. Many running related injuries are caused by doing too much too soon. The loads imposed by running are high and repeated. So you have to allow lots of time for adaptation. We have ad good success getting people back to running with a months long very gradual run-walk progression. It takes patience and discipline.
If you are so badly injured that this is not possible then hiking is my next best suggestion. No where near the stress loading of running but many of the benefits especially when a significant amount of vertical is included.
Cycling and swimming are both great exercises but their transfer to climbing big mountains will be limited once past a basic level. All the runners and XC skiers we train use swimming as a very effective recovery workout. I usually push 2-3 swim workouts a week for them. This really helps bring the legs back after long runs.
Hope this helps.