You are mixing two questions in one.... the maximum EV draw that panel will support, and the tie-into-old-breaker question.
Its not allowed to just tie into any of the breakers with a new 240V circuit(6-20 is a 240V circuit, maybe you meant 5-20). Assuming you meant 5-20... If something is already on a 20 amp breaker(especially in that panel), its probably because its expected to take 15-20 amps, and adding an EV which will take 16 amps continuously to that would be a guaranteed trip. Additionally, since you may be nearing the limit of the panel in 240V circuits any additional load on either 120V leg has a chance of tripping the main.
You(or your electrician) would have to do a load calculation to see what capacity is left in that panel. Its probably capable of taking a 30 amp circuit(24 amp EV load), but that depends on the ratings of the various items in play. Things like ACs are frequently over-breakered due to their motor-startup requirements, but other things like the dryer and stove are less so, but still sometimes have a decent amount of overhead. If you get model numbers(or nameplate ratings) of the big loads you can get a much better idea of what capacity is left.
In your position, I'd probably combine a couple lighting circuits(assuming you find the unlabeled breakers are actually in use!) and add a breaker to get 30 amps out to the subpanel in the garage, and figure out which type of breaker/outlet to put in the garage based on the aforementioned load calculation.