Could you import an illegal car, but keep it if you make it pass emissions tests?

Actually, the 25-year thing is a myth. ANY car you want to import from outside the USA MUST meet the safety and emissions standards imposed by the US Government on similar cars of the same year that were built for the US market. Just because the car is 25 years old DOES NOT mean that you can drive around with no seat belts or safety glass or spewing noxious fumes all over the place from an uncontrolled exhaust.

Yes and no. See the sources. You can import it but cannot register it in any state or drive it. You would have to get it certified (so that any car of that make, model, and year was certified) and that costs many millions of dollars... which is why the manufacturer didn't do it.

Are emissions the only reason for not being able to import one?

Since the engine electronics are so integrated into the car hood luck trying to swap the engine. All you have is a body without any befits if you manage it.

It won't even make it to the docks, since it is not 25 years or older.

You'd still have to pass the crash test, and guess what that involves?