What exactly would stop me from registering a domain from my own nameserver?
Nothing stops you from setting up such a nameserver.  However, this wouldn't be of any use to you, because no one is going to contact your nameserver to ask whether domainexample.com exists: they're only going to contact the official name server for .com.

It'd be like saying "can't I lie about the location of the White House if anyone asks me?" -- well, yes, I can, but if no one ever asks me that, then this isn't achieving much.

What prevents you from lying about .com?  People aren't going to ask you; they are going to contact the root nameserver and ask that question.  How do people know who the root nameservers are?  The root nameservers are publicly known (specified in a RFC) and hardcoded in everyone's DNS software.  So ultimately the recursion bottoms out.

That's why setting up your own nameserver won't help you: no one is ever going to ask your nameserver anything or have any trust in it, so its lies won't affect anyone.