You have rediscovered a very nice idea! This variant is known as no-pass-go (or sometimes Conway go, named after John Conway). It is very close to regular go, but there are a few differences. The main difference is the post-endgame stage you mention, but there are also some other side-effects, see for instance this post.
To turn your variant into basically normal go, only one additional rule is required:
- You may return one prisoner to the opponent instead of making a move.
(so essentially we re-introduce passing, but with a limit, making it so that someone will still run out of legal moves eventually)
With this rule, komi is nicely handled by simply giving white something like 6 or 7 extra prisoners before starting the game.
No-pass-go with prisoner return is my favorite rule-set for mathematical elegance, but there are some other nice candidates for beginner friendlyness. See for instance the rule-set used in this introduction, where the game ends after two consecutive passes, and each player’s score is simply the number of stones they have on the board. Which exact ruleset is easiest depends on the particular beginner, but these two (no-pass-go and stone counting) are both good options to keep in mind.