Back in the day, I used initiative to determine what order things happened, and that included movement. (I did individual initiative each round, since I wanted a little more tactical feel.) I did a simple d10 roll, modified for Dex (Reaction/Attacking Adjustment, with the sign flipped) and for weapon speed (Speed Factor from PHB p. 38, divided by 3, rounded to nearest integer). Negative numbers were possible (and such results were handled in sequence low to high), as were numbers over 10 (slow weapons could actually result in an attack being delayed into the next round at segment roll-10, which gave people an incentive to use lighter weapons that might not do as much damage). Spells were d10 + Dex mod, and began on the roll result and ended after a number of extra segments for casting time; long spells likewise could delay the final result into the next round.
As I recall, I used movement as a delay. When the number with your initiative came up, if you wanted to move and then attack, you could...it was a simple enough thing to figure out how far someone moved, and what fraction of a round that would take (based on racial movement speed, straight from the Monster Manual), and then tack that on to the initial number to represent the segment they'd attack on. (And if the movement took them into the next round, it was done just as noted above.) This allowed for some tactical situations to develop on the battlefield, with people moving to try to take advantage of positions to allow multiple-on-one or flanking maneuvers, though the monsters could do the same. The battlefield became a bit of a chessboard, and players actively looked for ways to anchor a position to prevent flanking (or take advantage of badly positioned opponents).
And yes, if you try to think of it as people "just standing around" then it doesn't work, but it isn't people "just standing around". If you read any books that describe a fight, you'll note feints, posturing, feeling out an opponent...all things which contribute to the final initiative result in game terms. The guy who got the better roll is the one who was first able to create an opportunity to make a meaningful attack. He wasn't just standing around, he was trying to make something happen. That's also a reason why you can consider Dex as an initiative modifier: faster reflexes means being able to take advantage of smaller windows of opportunity, and thus force a result more quickly.
Depending how much detail you want, you can add concepts from wargames (which in fact did get added in later versions of the game) such as zones of control and attacks of opportunity, so that someone can't just decide to waltz by a bunch of people without consequence. Or you can leave them out...which can make combat quite chaotic and fluid indeed, as people take advantage of any possible opening. The thing is, at this point we're branching deeply into house rules, and any answer becomes less authoritative and more a matter of the opinions and preferences of the people running the game.
If you do want a more theater-of-the-mind type of combat, then Seven's answer is pretty much the way to think of this, however. Discard the board-wargame tactics mindset, let the numbers dictate general order of action without getting all bothered about the detail, and concentrate instead on using description and improvisation to make things more interesting. If, on the other hand, you want a more tactical way of doing things that takes the game closer to later editions...well, then welcome to house-rules land, and be prepared to do a lot of tinkering to find the answers you like. What I outlined above is only a fraction of how complex you can make things if you really want to.