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.
Maybe you could just mention this briefly, as a sidenote, without theory. Something like: "Note that if we'd replace the numbers 1, 2, 3 with 1001, 1002, 1003, or 1000, 2000, 3000, nothing changes. (Show three versions of the same simple decision tree.)"
I made a series of game theory videos that carefully go through the mechanics of solving many different types of games. I optimized the videos for my future Smith College game theory students who will either miss a class, or get lost in class and want more examples. I emphasize clarity over excitement. I would be grateful for any feedback.