You are 100% allowed to join the DNC, run a campaign against Biden, and spend tons of time and money trying to convince the DNC to give you all their money instead of Biden for a presidential campaign. It won't work, but you can try.
Here's an article about 5 incumbent presidents that got primaried. Spoiler: they all won their primaries
https://time.com/5682760/incumbent-presidents-primary-challenges/
Well yeah, they already have an incumbent and no one was dumb enough to burn piles of money running a campaign to try and primary that candidate.
Yes, political parties are basically private clubs, you join them, donate money, pledge support etc. They aren't public entities any more than the Dallas Cowboys are. They exist to aggregate money and votes, and they elect their own leaders to manage those resources. If they decide to choose their candidate via tea leaves or throwing bones, no one could stop them, save to elect new party leaders.
I don't know how parties are run elsewhere, but thats how they work in the U.S. They aren't constitutionally codified or defended in any way. They're a shortcut for voters and super popular because they save people time and effort. If someone came up with a better way to all but guarantee votes for a candidate, it would instantly replace parties.