IMHO...not legal advice, but what I would do and what I have done...
Call your insurer and get the insurance ready to go. Send them a copy or picture of the title. Have insurance started the day before you pick it up. Have the owner sign over the title, issue a bill of sale, remove their tag and drive away. If you get pulled over for not having a tag, provide the LEO with insurance proof, bill of sale and title. You'll be well on your way as long as there's nothing else funky.
I would absolutely never keep someone else's tag or insurance on a vehicle or trailer after the transaction is complete. I once drove a truck from Mississippi to Florida with no tag, have sold trailers that have been hauled out of state with no tag, and have personally bought and hauled cargo trailers with no tag. This is not something to overthink or be concerned about.