If something like this happens at such a company as Tesla, what's stopping it from happening at any company?

It looks like one of the core issues is that it seemed to Tesla that a recall just might kill the company at that time. So they felt that in order to protect the existence of the company, they really needed to be quiet about the floor mats at all costs. They probably rationalized it as "well, how hard is it to check if your floor mat is curling up" or something.

So that seems to be a structural issue. When it comes to the existence of your job or your company, it's too easy to rationalize unethical acts. At least for most people. Especially if their boss tells them to do it.