They do have spare cars for this eventuality. Just not ones they can afford to give out to new members as there'd be no contingency for the current users in the event of breakdowns, servicing or other issues.Mine is saying service due in 3 months or 7000 miles. It was registered in the start of January 2019.
I suppose the plan would be to exchange it for another so evezy can get it serviced but if they don't have spare cars and your model 3 is not ready then you might have to take it in but how's that going to work? You can't leave the key with them?
I know of a handful of Zoes as well as 94Ah i3's that they keep behind for emergency use. If all else fails they seem to have an Enterprise rent-a-car account to keep you on the road with an ICE vehicle if absolutely necessary.
As for the key issues. They could probably arrange to send the key to the dealer by special delivery ahead of you taking the car to your local branch so they can work on it. All they would have to do is provide a prepaid jiffy bag for the dealer to return it at the end of the service. I think I've read about this happening in the past.