sure this would work, and the traffic would probably be not be that much. but as @Simon mentioned, this is centralized and can theoretically also be used for some sort of tracking.
but maybe all this is not really needed - already in the current version, we check if the apk was installed by f-droid or gplay - and if not, we show an annoying warning after a fixed timeout (90 days or so). at least this will mitigate the situation that many old versions hang around “forever”.
so, from this point of view, the problem is solved. and people who are interested in new features etc. will probably also follow other channels to get versions updates - this forum, blog, github etc.