The only way to beat this triviality is to mix in luck (where you have to learn to play probabilities) and psychology (where you have to learn to play The Man). This is why I find Backgammon to be a far more interesting game than Chess.
I find your opinion interesting, but disagree. Having spent a large chunk of my life playing strategy games of both the highly deterministic and partially random variety (let’s, for now, represent them with Chess and Bridge), I don’t find that luck alters that “perfect move” scenario at all, for me. The best players will usually play the same move when simple probability (with no psychology) intrudes, as in backgammon, the move that maximises their chance of winning. Much of the time it will work. Sometimes it will not. Therein lies a certain frustration; the strategy may be no less pure, but it ends up taking longer to distinguish an excellent player from a good player, and a good player from a merely okay one. The game isn’t rendered deeper by the added distraction of luck; it doesn’t make the game any better to the strategist, but only to the gambler.

Psychology is another matter, and one too easily lost in games of perfect knowledge. A game I used to play was both deterministic and relatively rich in statespace, but for each turn both players submitted their moves simultaneously, and only saw the results when both moves were in. For the most part, this game was susceptible to the same kind of thinking as something like chess; exposing yourself to less risk through more careful play was normally the best strategy, and within those who understood that basic axiom there came hierarchies of understanding what those risks were and how to defend against them. However, when two players with a roughly equal understanding of the strategies available clashed heads, there was room for a certain subtle psychological manoeuvring, a taking into account of the way the other had jumped in the past and attempting to take advantage of it.

However, even then, it was possible to banish psychology from the simplest levels of decision by carrying on with the same brutal logic of eliminating risk. To illustrate, suppose both players could choose A or B, and if both choose the same the first player loses; if they choose differently, the game continues. Either player could opt to eliminate the danger of being predicted by simply flipping a coin. And so, at the last, even psychology may fail to provide an edge against the truly random number.