It's Lazy but it's a 10X Stronger Body Position 

I would say you need the whole body training and strength. Without a core you can only play chess

It’s the grip strength and forearm fatigue that get me

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I think having strong legs is the most important.

Kettlebell training is a very good option for MTB. A lot of hinging motion, unilateral, requires a lot of stability to perform, complex and incorporates almost entire body, great cardiovascular benefits. If you find it "too easy" - try snatching 32 kg for 10 minutes like they do in competition, one of the best ways to do a whole body VO2max work. It won't yield a bodybuilder physique, but it is simple and time efficient way to train general athleticism with a single tool that can be stored anywhere.

I would say upper body pulling strength/grip strength has a bigger impact than upper body pushing strength. Stuff like rows/pull downs/pull ups and other exercises where you hold on. I ride BMX/MTB and started armwrestling which emphasizes pulling strength. The strength carry over was immense, I was able to pull the bike up and control it so much more easily, both on BMX and MTB.

What about for climbing up hill?