It's not up to them to "take" the car. It's up to you if you want to keep it or not. If you keep it, you keep it all. If you let them take it, they're taking it all, not less engine, interior, etc. It is challenging enough to gauge the salvage value of an older car as it is. Trying to do so without an engine, etc? They're probably not going to be inclined to invest any time in figuring what the car would be worth partially stripped. As I mentioned earlier, maybe they allow you to swap the wheels/tires out with other ones, but that's likely about it. You might end up getting lucky, but I wouldn't count on it.

There's no harm in having the other driver's company look at it and present an offer to you. Just remember, if their offer is less than what you would get from your insurance company, don't get mad, and don't stress about it. Just go back to your insurance company, and let them take over. Your Agreed Value coverage with your carrier is the perfect safety net.

I know an excellent personal injury attorney if you would like a no-string recommendation. You can PM me if you are interested. He was a litigation rep at a huge insurance company before moving into his own practice many years ago, so he knows the business inside and out.