Okay. Did the research I needed to do, I think.
I'm not sure I'd say you're placing "too high an emphasis on moving on your turn" so much as that the idea that there are "turns" feels highly disruptive to getting to the feel of a 2d fighter because of the interplay I mentioned -- if you are within your Move of your opponent on your turn, you are GUARANTEED to be able to reach them. And that's not how 2d fighters play out, generally -- unless both fighters really want to be at the same range, it's all about people trying to outguess and outplay each other to get to their preferred range. And just being able to "Move 3, then attack" doesn't mimic that at all, no matter where the sources of movement come from.
I think that this is a somewhat unique challenge to trying to emulate 2d Fighters because of what they look like. And the difference I'm seeing between the system you've laid out and the system of a 2d fighter is simple:
One character is a grappler and wants to be at grapple range. They have Move 1. The other character is a Zoner and wants to be at Long range. They have Move 2 but only when moving away. Right now, as written, it is functionally impossible for the grappler to EVER catch up? What ways are you giving them to try to outguess their opponent -- how do you simulate the "I think you're going to throw an attack RIGHT NOW that hits in a straight line, so I'm going to jump forward, even though that's an objectively terrible decision if you DON'T throw that attack" decision making? These contests are not, generally, decided by which character can move faster. How do you simulate "I'm going to wait for you to throw a projectile so I can move through it with my move that's immune to projectiles, and thereby get a little closer" in a turn-based setup? or "I think you're going to try to throw me, so I'm going to jump, even though that would leave me horribly open otherwise"? These are the kinds of decisions that I think are central to a 2d fighter, and they just... don't really work in a turn-based setup, because that "I think you're going to do X at this moment" kind of guesswork can't exist. Either you KNOW they just did X, or they will know on their turn that you just did your counter and there's no reason for them to then do X. Even a "reaction" doesn't really solve this, because if I have an out-of-turn reaction like "Block a projectile and advance one range band" I can use that with 100% reliability each time my opponent uses a projectile...so they just won't. Unless it's unreliable, but even then all you've done is reduce the guesswork to a "Do I think a 60% chance of hitting you is worth a 40% chance of getting countered?" game.
The way I've seen this tackled elsewhere (Check out Heaven/Hell for what seems like a pretty solid implementation) is double-blind simultaneous reveal play. I feel like there's probably not much place for a traditional initiative system -- the way I see these games play out there are three states: Either both characters are 'in neutral' and trying to gain an advantage, or one character or the other has the advantage, and the get to press it until the other character escapes/does a reversal/finds an opening to counter or something like that. And I think emulating that is an interesting challenge, and one that is going to be very hard to meet in a "I go, you go" kind of game format.