The The Oxford Companion to Chess has entries on only 700 named openings, and lists another 1327 opening variations in the index, and I wouldn't be surprised if someone out there had them all memorized. For an algorithm, however, storing that number of openings is trivial, and Chess algorithms traditionally made use of high-quality "game books" which are move-by-move records of chess games.
The problem with intractable models, like chess is that they are unsolvable, so brute-force is not an option, and any decision on a move, whether "memory" or "optimality" based, at least in the context of current advances, are probabilistic. (The exception is endgames where the the problem becomes tractable, and perfect play can be achieved.)
Essentially, the memory of a board state is only a small piece of the puzzle because the unsolvability of the game means that particular board state from a recorded game, where that state led to a successful outcome, will not guarantee a similar result in a new game, because new, more optimal strategies can always emerge.
Secondly, because of the extreme variation engendered by non-trivial games, it is extremely unlikely any current game would mirror the previous "memorized" game.
It is my understanding that "memory" as you define has been utilized in Chess engines, accessing a database of previous games to evaluate a position and derive a good move, but the intractability of Chess in general makes this method sub-optimal, and it has been definitively supplanted by newer methods (machine learning, neural networks).
In the famous Giraffe Chess paper, Matthew Lai states:
The job of the evaluation functions is to assign scores to positions statically (without looking ahead)
But the method is highly nuanced and probabilistic. Nevertheless, I think you'll be interested because the problem presented in the paper is rooted in the human vs. AI play method issue:
...the way computers play chess is very different from how humans play. While both humans and computers search ahead to predict how the game will go on,
humans are much more selective in which branches of the game tree to explore. Computers, on the other hand, rely on brute force to explore as many continuations as possible, even ones that will be
immediately thrown out by any skilled human... How can a human searching 3-5 positions per second be as strong as a computer searching 200
million positions per second? And is it possible to build even stronger chess computers than what
we have today, by making them more computationally efficient? Those are the questions this
project investigates.
There is no simple answer to the trade-off question due to the complexity of the problem. But this is an area where tremendous advances are being made.
I'd strongly recommend reading Giraffe Chess (September, 2015) as a primer.
It produced very exciting results, synopsized in the MIT Tech Review article "Deep Learning Machine Teaches Itself Chess in 72 Hours, Plays at International Master Level". (Giraffe's creator currently works as a Research Engineer in Machine Learning at DeepMind.)
