Up to D&D
I believe it is important for the classic D&D DM to move beyond a simplistic “A starts & completes action; B starts & completes action” model. The DM must visualize the situation & determine what happens based on that rather than trying to slavishly follow the book’s procedures.
Actions are overlapping. Initiative determines who has the edge.
e.g. Two groups encounter each other & close to melee combat.
What shouldn’t happen: The side that wins initiative crosses the intervening space while the other side stands still.
What should happen: Both sides meet in the middle & initiative determines which side’s attacks are resolved first.
Likewise, if two characters are racing to grab an object equidistance between them, the one that wins initiative will get there & be able to grab the object just a moment before the other.
Another example: A fighter is moving cautiously through the wilderness. He has his bow strung & an arrow nocked. He suddenly encounters the goblin. Neither is surprised. They are close enough that the goblin can close the distance & attack this round. The fighter declares that he’s going to shoot his bow. The goblin chooses to close the distance & engage in melee.
No initiative roll is necessary. The fighter can get his shot off before the goblin can close the distance, so the outcome would be the same no matter who won initiative.
Now, if the goblin choose to dive for cover instead, initiative would determine whether he’d get the protection from cover this round.
I have found that doing this often relies on the declaration phase. (Which is a topic of its own.) I have to get an idea of what everyone is doing before I can visualize what is happening. This has two drawbacks.
Firstly, it slows things down. Secondly, it isn’t scalable. As you get more players, it is harder to keep what everyone is doing in your head.
I have found that, in practice, I often end up resolving things in this order:
Missile attacks go first, because missile typically move faster than the combatants. Movement comes next, because missiles and movement typically cover more distance than melee. Magic comes last because I have spells go off at the end of the round. (Casting a spell takes more time than the other actions.)
So, these days I tend to use this structure, although everything above still applies.