I think the controller is a big part of it, but at this point there’s also a different built in player base for strategy games and consoles. RTS definitely tend to have issues without keyboard and mouse, but you don’t see turn-based strategy as much on consoles as on PCs either, and the controller is less of an obstacle in that case. I think it's mostly controller and economic factors that cause it, but there's a perception that strategy games are something you play on your PC, not in the living room. Economically, I think it ends up being a problem with limited player bases (limited to everyone who bought that console) that make game companies want to avoid the less accessible genres for the fear that they won’t be able to sell enough copies to break even, especially for exclusives. Even with something like the Wii U controller, which is well suited for certain kinds of strategy games, there’d probably be a relatively small percentage of the already limited set of console owners that’d even be interested in them.
As for MOBAs, they could be successful on consoles, but I wouldn’t classify a MOBA as a strategy game any more than I’d classify a team shooter as a strategy game. There’s an element of strategy in most games, but they don’t have much in common with traditional strategy games other than the fact that DotA was originally made in an RTS game’s engine.
MOBAs enjoy mainstream success because the core gameplay is designed to be accessible and encourages constant action, while traditional strategy games are less accessible, which is why they’re more of a niche genre and therefore less suitable for consoles. MOBA isn't my kind of genre, but it has s a very successful design philosophy for a few reasons. The towers, the creeps, and the leveling system in MOBAs are all meant to ensure that the match lasts a decent amount of time (without turning into a complete stalemate) even if the teams are mismatched in terms of skill level. This works at the casual level because it ensures a minimum playtime, so even a fairly bad team has a chance to play and get practice without being rushed out of the game in the first few minutes.
At the same time, the attrition style of MOBAs also works at a professional esports level because it reduces uncertainty and volatility. Not only does this lead to longer games for viewers, but it makes it less likely that a worse team will beat a better one. It’s the same concept as the law of large numbers, if you think you're the better player, you don’t want the game to be decided on one early fight, but by a lot of fights throughout the game. When the game makes it harder for an unscouted rush to end the game early on, that gives more time for the better team to gain an advantage. That’s why SC2 pros also tend to view long macro games as more legitimate than cheese (rushes) because a less experienced player can still win a few early engagements and take a game, but that same player just won’t out-macro a full time pro without putting in a similar amount of practice.
