I've mentioned this as an aside once before, but here's how I do initiative. Once I explain how sublime and wonderful my system is, I expect it to catch on like wildfire throughout the OSR. Or, one person might cotton to it and end up using it with a tweak or two. It could happen.
All the players know their weapon speed. Yeah, that table on p. 38 of the Players Handbook? I use it. Monsters with natural attacks (like claws or bites) add zero.
Anyone who's surprised automatically loses initiative. Duh.
When the round begins I say "New round, new initiative!" Everybody rolls a d10 every round. I roll one collectively for all the monsters. It's just easier.
The players add their weapon speed or spell casting time (in segments) to whatever they roll, and subtract their reaction/attacking adjustment (for Dexterity). If they're doing something wacky like swinging from a chandellier and leaping on the foe, I just make up a number and tell them to add it to their roll, because I'm the DM, and that's part of my job description.
I start counting off. 1... 2... 3... 4... When I hit a number of a player's modified initiative, they say "I go!" If I reach the monsters' number, I say "They go".
And that's pretty much it.
You roll a die, add one number, subtract another, and BOOM! You're done. And since they don't change unless you change weapons, it's pretty much a set modifier. It is really that simple.
There are a few special cases, of course, like missile weapons. Godsdamn missile weapons. Look at the rate of fire on p. 38 (ah, my beloved p. 38). If it's 1, you go when you rolled. If it's 2, you go once when you rolled and again at the end. And so forth. If it's 1/2, you go every other round. Simple. There are other special cases, but that's the point of being a DM. Make it work; it's easy.
I have to say, this has worked so very well for me over the years I wanted to share. It gives a benefit to those who choose swift but light weapons over bulky but punishing weapons. It makes casting times meaningful. It gives all the flexibility a DM could ask for.
Give it a try!