I was an alpinist and ice climber for ~15 years, and I’ve been skimo racing for five. Skimo will give you much better sport-specific support for running than climbing.
Walking around with heavy packs is much less specific than moving on light skis with a (super) light pack. Plus, the bang-for-the-buck factor is pretty low in climbing. When I go climbing now, I’m amazed at how long it takes to do so little.
I love climbing, but if running was my priority, I’d focus on something more similar in the winter.
I hope that helps.