What actually makes a game "deep" and interesting.
Let us first look at Rock Paper Scissors. It is a simple guessing game where you pick from three options and you're either right or wrong. There really isn't much to learn, as long as you understand those rules you're set for life. Despite that, some people can have a lot of fun playing RPS. Is it really deep? Probably not, no, but that isn't to say it's not fun given the right situation.
What about chess? Chess is technically a "solved" game in the sense that a computer programmed with the absolute optimal plays should win every time. But most of us are not computers, so the game is interesting to us because we have to learn these optimal plays, memorize them, and then exercise our ability to remember. As in my last post, when the brain fails to remember it's human nature to guess in order to "fill in the blank" and hope for a correct answer, which can also be interesting to us.
When a game is complicated enough that humans can not fully master it you can say that the game is "deep" because we are always in pursuit of those optimal plays -- but achieving 100% optimal is just not humanly possible in some games. Just like how chess is "fun and deep" even though it's technically solved, because humans can't play at that capacity.
Adding RPS elements to a game doesn't actually add depth though. All it really does is give the illusion to weaker players that they could possibly win if they get lucky, and aggravates skilled players when they are unlucky. In the end, RPS may look like depth, but it isn't.
Fighting games have the issue of entertaining both elements marginally. When you speak to the old veteran players they often talk highly about exploring the game in order to discover the optimal plays. The "glory days" of fighting games are often when a player discovers a strategy that is seemingly not possible to counter, then some one revolutionizes the situation by coming up with a counter-strategy. Just like with chess, they are seeking to solve the game.
You could argue that fighting games are significantly less deep than chess because it doesn't take us nearly as long to solve a fighting game as it did with chess. Plus, even once fighters/chess become solved there are drastically fewer people who can play on an optimal level in chess by comparison to how many players can play optimally in fighting games. A great example of this is to just look and see how many Yun, Chun, and Ken players all play exactly the same (optimally). What separates them is not their mental ability to remember optimal plays or decision making, it has more to do with reaction speed, guessing/luck, and execution: physical abilities.
Compare for a moment a tool assisted play of both chess and a fighting game. Both players are playing at totally optimal levels in both games. The object for the spectators would then be to try and recall what the best possible options for each situation is as the games progress turn-based style. Most of us would probably choose an incorrect answer in chess fairly easily, but that might not be the case in the fighting game..
So why do people keep playing them once they are solved? Once a fighting game's optimal plays are solved it reverts back to a question of reaction speed, execution, and guessing. In other words; after a certain number of years it's more of an exercise of our physical abilities rather than our mental ones, and RPS/luck becomes more emphasized. But just like with RPS they can be fun in certain situations.
After all, we still highly enjoy most mainstream sports, even though the difference between most players is physical ability and not mental ability. Surely there isn't much to learn for a game like baseball, but we tend to idolize good baseball players purely on their physical ability.
So, the counter argument is that depth is in itself the game's capacity to test our ability to play at optimal level, be it mentally or physically (or both). It's fair to say that you shouldn't add RPS elements to a game because it's neither a test of physical or mental ability, it is only an exercise of luck which both players and spectators generally find unappealing.
With fighting games (and RTS) I think veterans appreciated the mechanics of the game more. And in the end the audience appreciates physical ability more. The suboptimal "flashy" eccentric plays they will win you the audience, but not always win you the game.