They probably would. If the Saturn is older than 25 years, it can be registered as a classic car. The fact that it's highly modified with new parts doesn't really matter. It's what people in the hot rod scene have been doing forever.

For newer cars, you could probably register it as a self-built (kit car).

The states register vehicles, not insurance companies.

And while a big box insurance company might not insure a heavily modified vehicle, there are niche insurance companies who will. Or you could even self insure in a state that allows it.

Why not? The only issue is if that Saturn was scrapped - once a car is scrapped there is no legal way to get it titled. (but you can still call it home built with parts from the scrapped car - it just needs a new VIN).