What a great question.
One important mathematical model for chess is Combinatorial Game Theory. Under the vanilla version of this beautiful theory, developed by Conway, Berlekamp & Guy, one loses when one has no legal moves. One modification of the theory is so-called "misère" (a term taken from card-games) where one wins when one has no legal moves.
A game and its misère form thus share the same game tree. They just have opposite values assigned to the leaf nodes. One can derive values for all other nodes in the game trees by working back from the leaf nodes. One amazing feature is that the value of non-leaf positions between misere and non-misère versions of the same game can be very similar.
For example, the winning strategies for the well-known game of nim are essentially the same. It's only with the winning player's final choice that they decide to take one more or one less match from the last non-singleton pile.
Another way to say this: players vie for strategic dominance. Once that is achieved, then one could force the opponent to do a number of different tasks: be checkmated, to checkmate, to send king on journey to occupy every square of the board, etc. The difference between these tasks would be secondary - once strategic dominance is achieved. In practice, it's not as clear cut: part of achieving strategic dominance in chess involves threats of mating. An attack may sacrifice strategic dominance for a winning tempo advantage in order to checkmate quickly. But nevertheless, regular chess and misère have much in common.
Now chess problemists have long had a misère genre they term "selfmates". (The problem database PDB contains over 58,000 of these today! Type g='s#' in the window https://pdb.dieschwalbe.de) These are about the dynamics of forcing a nearly-dominated opponent to execute the victory task, in an interesting way. They tend to involve long series of checks (so-called entailing moves in combinatorial game theory). For example https://pdb.dieschwalbe.de/P1014331 takes 10 moves. All this means that the initial game array is unpromising as a start for a selfmate. If I had to bet (and of course we can never know), I would hazard that misère chess is a positional draw from the starting position. I think this is true even if it turns out that orthodox chess is a win for White, because in misère, tempo does not have the same value.