There is plenty of research on this, but it often goes on under the auspices of "artificial intelligence" rather than "game theory". For example, last i knew, the best algorithm for poker was called "deep counterfactual regret minimization", which as the name implies is an implementation of counterfactual regret minimization that fits a parametrized deep learning model. The explicit goal of that algorithm is computing approximate nash equilibria, it just wasn't developed by game theorists.
It is interesting that work like that tends to be done by computer scientists rather than by game theorists, and i think the reason for that has to do with the respective incentives and intellectual tools. As an outsider looking at the literature, it seems to me like game theorists think that writing proofs is what their peers want from them, and they don't seem to have any expertise or interest in things like practical function fitting or equation solving.
Computer scientists, on the other hand, can win a lot of recognition for themselves for practical accomplishments, such as creating and implementing algorithms that actually play games, and many of them have the expertise necessary to actually do so. They naturally have the means and the opportunity to create parametrized algorithms.
