I think the reason the 1-up mushroom doesn't give you a "power" like traditional mushrooms, squirrel mushrooms, etc. is because it gives you a one-time ability to perform incredible feats of strength, like come back from near-death or do a super jump out of a pit or lava (this is why Mario can double jump off of air in Smash Bros., it's a weaker version of the same power). I don't think the 1-up mushroom itself brings Mario back to life, or else the Life Shrooms in Paper Mario would be useless. In M&LRPG one-up mushrooms are used similarly, but you'll notice one of the brothers has to force-feed the other, rather than the full resurrection being intrinsic to the 'shroom.
In Wario's case, the same principles largely apply, but Wario is just generally stronger than the man in red. How many Wario games could you even fall down pits in? I'm thinking WL1, SM64DS, World, and Shake It? When it's convenient, Wario just expends some of his Garlic-Powered Strength to hop back out, but most of the time he prefers to explore the depths for even more treasure! There's even a minigame in the first WarioWare that makes reference to this (you have to steer wario into a hole, so he can find a WL4-esque diamond).