Do you know what the primary, most important, difference is between old school style and new school style games?
When I returned from Alaska, I started a Hackmaster game with friends. The real reason that happened, is because monsters in 3.5 didn't have a morale score. I was trying to run games play by e-mail from Alaska, and was looking for an unbiased metric for monster behavior. I even asked about morale (and got answered) in sage advice.
So, practically, I went with Hackmaster because I had it, and it had morale values. The more I learned about Hackmaster, the more I saw the way it elegantly handled a number of issues that came up in first edition, such as mages using armor, lethality, character behavior, et al.
But it used d10, second edition style initiative and I didn't like it.
There were a couple of things that were troublesome. Any positive factors and you could lose your turn (You got a 14, you go next round!) which was a problem in actual table top play.
So we tried everything. Everything mechanical, at any rate. We tried a d12. We tried a count system. We tried a step system. We rolled multiple dice and had rolling, continuous initiative. We tried declare first and roll. We tried, well, a lot.
You know what worked best in play? Vegas Style 1d6, roll then declare initiative.
Oh, there's so much wrong with it! It doesn't track weapon speeds! It doesn't allow interrupting spellcasters! It doesn't allow individual character factors to modify your initiative! It's all that wasted design space!
I'm a designer. I want choices to matter and have an effect in play. Initiative is a great place for that!
Only it isn't.
Lots of things differentiate the different play styles. But the most crucial is speed of combat. Yes. Player skill this, character builds that. But in the heat of actual game play, having a combat be fast and deadly, allowing you to spend more time on exploration, is the core of old school play.
And having individual initiative removes that.
The instant you make each individual take their own turn in order, you turn a five minute combat into a thirty minute combat. You use a map with letters and positioning, you double that again.
And after playing Dungeons & Dragons for thirty years with hundreds of people: Having a six hour session where you have five hours of combat, versus one hour of combat, has been the most obvious and pointed difference between the styles of play.
You don't need initiative.
It wasn't even in the game until Eldritch Wizardry was released. Before that the only determining factor was weapon length.
All games will work perfectly fine (and very fast) if you just let the players go, and then the monsters go.
But sometimes, it can be useful. You don't want every combat to take an hour, but you might want the important ones to! This is even touched on in Advanced Dungeons & Dragons where it's suggested using a more complicated system for one to one duels.
Some of it is simply a matter of taste.