One could reasonably propose that things ought to work that way, but that is not, in fact, the way things have ever worked, at least not in the USA. People routinely make very substantial alterations to their cars and continue driving them on the street with no legal process involved. "Street legal" is not an exacting standard and no inspections are required for an altered vehicle (so far as I know, anyway) beyond whatever routine process your state requires as part of the registration process. (Here in Washington state, that's just an emissions test.)
I used to be part of a 4x4 club; we routinely rebuilt our trucks' suspensions, changed out the gearing, replaced wheels and tires with bigger ones, installed power steering systems adapted from some other type of vehicle, swapped our engines or axles out for bigger, tougher ones, swapped out entire drivetrains - you name the component, somebody in the club probably modified or replaced it.
We weren't even doing anything unusual by recreational offroading standards, much less hot-rodding standards. We weren't building tube frames, and most of us didn't modify our frames at all. Nobody had ripped off all the sheet metal on their truck and replaced it all with a molded fiberglass replica; none of us installed superchargers or nitrous injection or anything like that. I'd guess no more than half of us even knew how to weld.
Modified cars are on the road all the time; you've seen thousands of them whether you realize it or not. If this were a big safety problem, there would have been a moral panic about it decades ago. Tweaking the software in a Tesla is completely insignificant by comparison. No, it doesn't need to be checked out and approved.
