OP is right. If two people are picking at the same time, neither one has pick priority. People seem to think it does, but it doesn't. It comes down to whoever calls what. Pick priority is only different if someone is picking at a different time than you.
If Riot wanted it to work this way, they would change the way it's done by doing this.
Team 1 (first position) picks. Team 2 (first position) picks. Team 2 (second position) picks. Team 1 (second position) picks. Team 2 (third position) picks. Etc.
Now, however, it's:
Team 1 (first position) picks. Team 2 (first and second position) picks.
The intention isn't even that pick order matters. It's simply a way to balance out champion selection. The intention is not to say that first pick should get whoever they want. Remember, it's a team game. You should work as a team. First pick should work with their team to make sure that the guy that's really bad at jungle shouldn't end up jungling. That's the real problem with this mode. Just because your first pick doesn't mean you should just lock whoever you want. You should use this opportunity to talk with your team to figure out what's best for the team...not just yourself.
OP is correct in saying that if you pick at the same time, neither has priority. However, that's not even the point of draft pick. The point is so that champion selection is even among the teams. The "pick priority" was invented by players later.
A few years ago, in draft, people called like they did in blind. Whoever called mid took mid. Doesn't matter what pick spot they were, and most of the time, everyone respected it.