Having played lots and lots of 2e back in college, this is how it was done (and with spell speeds as well, which really ratcheted up the tension if you were a spellcaster!). It definitely made the game a lot more interesting. The problem with doing it with 5e is that 2e didn't have anywhere near as many options for characters, so players were able to decide upon their actions much more quickly, so re-rolling initiative and adding weapon/spell speeds didn't slow the game down as much as it would now. Also, the groups I played in were pretty veteran - as they would be since the various friend groups I was in had like a half-dozen campaigns going on at any given time (it was the late '80s/early '90s, so no cellphones and very rudimentary internet that was hard to access and slow; so alcohol, relationships, and D&D were the main ways to fill up the time lol), so we all had the rhythm of combat down to an art. In a way, I do kind of miss it, but, thinking it over, again, it just wouldn't work very well these days.