I wrote a PhD which discussed, inter alia, what it means to make a promise.Yes, this is indeed what I said. I'd be interested to hear your reflections on what it means to make a promise.
I teach and study private law which includes, inter alia, considering what it means to make a promise.
The most recent piece that I had accepted in a major scholarly journal had a footnote discussing some aspects of what it means to make a promise (with reference to one of the important figures in the field, John Rawls).
But there is no need to enter in the theory of promise-making, or the discussion of whether and when promises are binding, in order to discuss how RPGs work.
Here's why:
Suppose that promises are not binding. Then they are not ways of creating or imposing rules, and hence we can set them to one side.
Suppose that promises are binding. And suppose that I promise to X not to stick to the written rules of a game, without changing them. Here are three possibilities that I believe cover the field.
X is someone who has no connection to the group of people with whom I am playing (eg I made the promise to my dying aunt on her deathbed). It then has no binding effect on anyone in the group but me. If others choose to play by the rules out of respect for my promise to X, that's a choice they make moment-to-moment. Promising makes no difference here except to reinforce the weight some might accord to my desire to play by the rules as written.
X is one of my fellow players, and we are playing in a social/friendly context: X can release from my promise at any time, and hence the promise has no effect on what rules are binding.
X is one of my fellow players, and we are playing in an institutional context such as a tournament: my promise is part of the institutional infrastructure, and it is that infrastructure that does the work of establishing which rules are binding (as per my post not far upthread).
In my view, the preceding disposes of the relationship between promise-making and RPG play.