Not only OSR games suffer from the same old, same old problem with initiative rules. Essentially, those rules boil down to two alternatives: the whole group gains initiative and the individual members can determine in which order they act, or the rules determine who acts when.
It works, of course.
(And personally, I like group initiative a lot.)
But I think the single best twist on these age-old rules are Dan Sell’s Troika! initiative rules. You can use them in every roleplaying game. Speaking from a professional standpoint (I’m a certified intructor for reality-based self-defense and instructor-in-training for a Russian martial art), Dan’s initiative rules are realistic. Realistic, as in yes, it really works like that in melee, it’s all a huge fucking mess, and things happen you just don’t want to happen and everything is going in all directions all at once.
So, yes. Do yourself a favor and use Dan’s rules. They’re that good. AND they’re fun.