Liability is definitely covered. It's not legal to have no liability coverage. Liability part is provided by a traditional insurance company that Tesla uses.
You can add spouse or any other person to your insurance as well, same as every other insurance I have used.
What’s the “national insurer” you keep talking about? There is a difference between Mercury and Farmers, but both are national.
Take a look at this thread. Many posts echoing my concerns. No proven track record. Insure at your own risk to save a few bucks.
Tesla Insurance