Poker at the very least is played for EV and moves are made with the expectation of the randomness converging over many samples. This is partly why tournaments are considered luckfests by most money game players since you need to win so many coinflips to win.
Well nothing prevents the game designer from implementing similar mechanics (lots of smallish randomized events in each game which will lead to variance reduction)
There is probably an interesting distinction to be made between the type of randomness that comes from a deck draw versus the type that comes from a loot table... but I don't know it. It feels different, but I can't quite put my finger on what it is.
I'm not convinced of that. In either case, the random element just makes the game tree larger.
What I like about randomness, is that adding it makes each of the individual games a unique story, which can have a great appeal (just look at the success of MtG,or Binding of Isaac for example). Obviously, designing such games (so that they're balanced) can be an order of magnitude harder than designing a game without randomness.