You are right that the usual assumption in game theory is that payoffs are ordinal. When I teach game theory I find it useful to mostly ignore this fact because some of my students (those who took Intermediate micro) have spent a lot of time on ordinal utility while most of the class has never encounter the concept before. In my video lectures I ignore the ordinal assumption except for assuming that players seek to maximize their expected payoff. (This follows from ordinal utility.) But all of the solutions I give would still be correct if you interpret the payoffs as ordinal.