It's a special case. You can't just allow everyone to go whenever as that breaks combat and makes combat unplayable, so an ability check is necessary. Specific beats general.
As @
pemerton has already pointed out, initiative doesn’t have to work the way it does in 5E. You could have all the action in a round resolve simultaneously, or use side initiative. Another suggestion I’ve seen made on these boards is to forego rolling initiative at the beginning of combat, waiting until conflicts in timing arise in the course of events and resolving each of those conflicts with an opposed DEX check. Clearly, turn-based initiative is a deliberate design choice in 5E, not a necessary one.
Also, the way the game has chosen to keep each character’s turn separate is to call into question the certainty of when each character’s turn will happen. It resolves that uncertainty with a DEX check, which measures a character’s ability to move and act quickly and is compared with the other results to establish a ranking. None of this is necessary for turn-based initiative. Each player could make an unmodified roll or draw straws if the goal is only to establish a turn order. The choice of a DEX check implies that the participants are attempting to move and act quickly and that they may fail to do so. Saying that it's a special case or that it's necessary doesn't explain that away.
I have. I was a loner in junior high and high school, but I fought back against bullies so I got into multiple fights a year. Had I been in school in the modern era, I'd probably have been arrested and expelled due to stupid changes in how they treat kids. I've been in enough fights to know that you are just trying to win the fight, not see if you can go first in the round.
I doubt you were fighting in rounds. Seriously, though, I don't see why "trying to win" doesn't entail trying to strike the first blow. Just letting your opponent hit you first seems like a good way to get knocked out.
That's wrong, though. An arm wrestling match is direct opposition. A Jeopardy question is direct opposition. Only one can get it right. Rolling initiative to see when in the round you can act isn't in direct opposition to anyone. You are not trying to stop them acting with the initiative roll.
You're trying to move and/or act before they do. They are trying to do the same. Those two efforts are in direct opposition to each other. Only one can succeed in being first. No one is claiming you're trying to stop them from acting at all. That would require incapacitating them in some way, which I think is beyond what initiative is meant to decide.
Specific beats general. Combat requires a mechanic to see in what order those who are automatically successful at being able to take an action can take that action.
I'm not sure what you mean by "Combat requires". There are any number of ways a combat system can organize the action. If the intent of the initiative phase is only to impose an order of resolution then why not flip a coin? Why use a DEX check at all?
There isn't a specific rule, though, when it comes to DM vs. player ties. The rule given is identical to what would happen if no rule existed. The DM would decide. It's a non-rule rule.
The DM could decide both turns resolve simultaneously, but the rule is clearly designed to avoid that situation.
