What happens if I let the registration lapse on a car I haven't seen in months?

Your partner was lucky. I am glad you are taking care of this. If the buyer won’t take it seriously and in an adult manner you need to go get the car. Bring your title and some liability insurance and drive the car home. PITA? Sure. But it probably needs to get done, soon.

I am not understanding how a non-owner of a motor vehicle can have insurance on it without some formal agreement signed such as a lease. I was told by a former insurance agent that you can only insure things you own or those of your dependents or others that live in your residence. My adult children still at home are under my policy to acquire the multi-vehicle discount and umbrella protections. Their driving records apply whether they are under my policy or not.

You aren’t missing some magic phrasing- you haven’t transferred ownership of the car until you’ve transferred ownership of the car. Which means that you’ve signed over the title to him. Now, there may be some way for you to transfer the title to him and have the transfer recorded without making certain that he registers and insures it. In NY, that would be a “Title only transaction” but that still requires him to sign your title and the application for a new title.

Remember, we’re talking about the car belonging to the OP’s partner, not the car the the OP sold. It’s possible that there was such an agreement. My dad once sold a car to a neighbor, and in the written contract there was a clause requiring the neighbor to carry collision insurance with loss payable to my dad.