I've been to the Cuban interest section in Washington DC several times. It's not really an embassy in that diplomatic functions and freedoms don't exist like proper embassies.
The ambassador's freedoms are quite limited. Eg, at some point he was not allowed to leave the confines of the DC beltway. Diplomatic immunity be damned. Really sucked for us planning an event in my Baltimore uni for him to speak at, where the travel restriction came into effect just days before and we had to resort to video conf.
Well, that's the thing: since it's not an embassy, it doesn't have an ambassador with diplomatic privileges, it has a "chief".
But my point is that this is all just paper. Out in the real world, the US can (and probably will) pretty much just swap out the plaque outside the "US Interests Section" with one that says "US Embassy", with the chief and their assistants magically transformed into diplomats.
The ambassador's freedoms are quite limited. Eg, at some point he was not allowed to leave the confines of the DC beltway. Diplomatic immunity be damned. Really sucked for us planning an event in my Baltimore uni for him to speak at, where the travel restriction came into effect just days before and we had to resort to video conf.