I quite like the idea but I played 1e for 20+ years where everyone but fighters only got to do one thing per round (plus a little shuffling).
This was the way we dealt with multiple attacks, and I think it would work for your suggestion. Basically, attacks are equally spread throughout the turn:
Say your Initiative was 21.
If you had 2 attacks they would happen in 21 and 11, 3 attacks = 21, 14, 7.
A Flurrying Monk with 5 attacks = 21, 17, 13, 9, 5.
2 attacks when you rolled 1 = 1 attack.
Works fine as long as your players are capable of basic arithmetic.