There are two methods of thinking when playing chess (or any abstract strategy game) -- calculational or tactical thinking, and heuristic or strategic thinking. In chess we might also call the latter positional thinking.
This is somewhat strange. After all, chess is a combinatorial game of perfect information. If you lose, it is not because you rolled a dice imperfectly, nor because you had to guess what an opponent would play and by misfortune guessed poorly -- it is because you played a bad move. Therefore, it is strange (at least to me) that heuristics should be involved at all.
Here's it put another way. When you play chess, you constantly make decisions based on probability. When you decide to put a knight on a central square, it's not because you know that 21 moves down the line you're going to be able to threaten a fork. You put that knight there because you know that from a central square, you're more likely to be able to get it involved in a tactic down the line.
We are forced to make probabilistic decisions because the search space is far too large due to the combinatorial nature of the game, despite there being absolutely no probability whatsoever in the game itself. From the moment you set up the pieces on the board, the game is one single extended tactical combination, one long forced win or draw; and the only reason it is ever not that is because at some point a player makes a mistake.
You might think chess engines are different, but in fact they're not! Sure, chess engines are able to calculate much, much deeper than humans, and that is the main reason why they absolutely dominate human players; however, they cannot calculate all the way down to checkmate from a complicated position (and likely will never be able to barring massive improvements in computer technology or the development of novel algorithms for reducing the search space).
Instead they look a couple of dozen moves deep (at most), and then evaluate the resulting position. And that evaluation function must be supplied by a human, based on centuries of human chess position evaluation heuristics. I mean, sure, a lot of it is common sense ("more pieces = more win"). But because of this, chess engines often overlook fortresses and other clearly drawn situations, insisting they might be able to squeeze out a win due to superior material. In this sense, chess engines also have no choice but to rely on probabilistic reasoning.
I'm not sure how exactly to make rigorous, mathematical sense of this paradox. How can a game without any probability require probabilistic reasoning? I'm sure some mathematician specialising in game theory will have done research on this. But I'll throw out an idea of how you might approach it.
The idea is to borrow a page out of statistical mechanics and the notion of microstate vs macrostate. You would divide the space of possible chess positions (microstates) into parts (macrostates) according to various properties of the position (such as material, piece activity, number of passed pawns, and such). You would then argue that within each of these classes, a certain fraction of the positions are won for White, a certain fraction are won for Black, and a certain fraction are drawn. Evaluation of a position then comes down to identifying the properties of the position and arguing that because the position has this or that set of properties, it is more or less likely that White or Black is winning or that the game is drawn.
The issue with this is that it just pushes the problem down one step. How do you figure out those fractions without just calculating every position? What our human brains do is they look back on all the games that we have played as well as all the theory we have studied, and assign approximate probabilities to these different macrostates. You could do the same thing: look at a database of all games, look at every position that arises in those games, classify them according to the properties you're studying, and derive win/draw percentages from that.
The issue with that is that it effectively introduces a notion of "metagame". It's conceivable that human play may get stuck in a metastable equilibrium, and so that your win/draw proportions are not actually objective but depend on the particularities of the metagame. Consider this as an illustration: if you fed this model games exclusively from the 19th century, versus games exclusively from the 21st, the former is likely to put much less stock on material and much more on piece activity; this is not because the game has actually changed since, but because of the style of play that was prevalent at the time (i.e.: the metagame).
This "macrostate" idea is arguably quite close to the way actual humans evaluate chess positions. We make arguments as to why White or Black is winning in this or that position, and then to support those arguments we point to reasons. Such reasons (such as "Black's king is exposed" or "White has a great outpost on f6") are the macrostates.
One last elaboration. Instead of considering only the ultimate evaluation of the game based on a property of the current position, you could approach this more "dynamically" by considering the probabilities of transitioning to other positions with different properties. For example, a material imbalance of 20 to 21 is quite likely to eventually transition to one of 3 to 4, while it's much less likely to end up at 19 to 12.