But, to me, this makes complete sense.
Imagine I'm on the street and a guy with his fists up is moving menacingly towards me. Depending on the distance, I have a couple seconds or so to react -- to withdraw, kick him, etc.
He knows that too, and will account for those possibilities. He moves in not exactly knowing how I'd counterreact.
If I was not paying attention or surprised (not in 'combat mode'), then feasibly, he could move into melee and then get his next action before I could react.
Ranged attacks ignore this complication but have different issues, thus making their combat tactics feel genuinely different, and not the same.
And this scenario gives fighters a reason to throw a spear before closing into melee, like they sometimes did in 1E (but have little mechanical reason to do so now).
Reach weapons also become tactically important.
The problem is when the movement occurs in a phased rules system. Let's say that you and I are 90 feet away. You have a halberd. I have a dagger. We can both cover 30 feet in a move. Both wanting to engage, we both move forward on our single action. Now, with just us involved, that's no issue at all. (There is the issue of how fast we can do this closing on each other while other people are, say, firing arrows are casting spells.)
But once you get into close enough range for it to matter, people don't typically run right up (sans charge). That is, I'm not going to spend my action walking right into your halberd range. If I move, I'm going to move about 20 feet, and start looking for openings. And more critically, I will
never stop moving. If I do, I'm dead. You will be doing the same thing, albeit your movement will be a bit different, since you want to keep me back and I want to get inside your reach.
Again, if we want to play that out one-on-one, the single action can work. The combat will take awhile, because we will spend a lot of those actions doing nothing but moving. But as soon as you mix bows, spells, and other stuff in, it becomes misleading. Now, maybe if you want to make melee that realistic, you would also make drawing an arrow, notching it, pulling it back, acquiring the target, and firing--take more than one action. And you'd set up most powerful magic to be the same way--gathering the energy, shaping it, launching it, etc. However, D&D has traditionally ignored all those details and/or abstracted them (aka, the 1 minute rounds of early D&D).
Once you abstract it somewhat, then my initial example makes more sense. We both want to close. We are both wary. We start 90 feet apart. We both "move" rapidly until we are within about 15 feet, and then we start circling and dancing around looking for an opening. (Or, if I'm not an idiot or overconfident, and we are equal skill, I'm running--knowing that you don't bring a dagger to a halberd fight. As initially stated, I must think I have some other edge to seek this engagement.
)
For a single-action economy that was less abstract I'd probably acknowledge that not all movement is created equal. Make each characters' "speed" higher than normal, but only let it apply out of melee when not doing anything else. Thus, we can "close" on each other rapidly (in number of turns), and it won't matter that bows and spells are flying equally rapdily. However, once someone gets into melee (or starts firing or casting or doing anything else), they are still presumed to have some minor tactical movement. The 5-step is insufficient, but 15 feet or so would work. (You can still have interrupts for those directly engaged to discourage abuse.) This latter movement would
not adjust by speed, because it is constrained by so many factors (armor, weapon ability, speed, dexterity, etc.) that you might as well call them a wash. There might be an option to forgo your "melee movement" in order to stay on a target--essentially moving with them without the hassle of an interrupt mechanic.
