I worry about this all the time. OPs experience is worrying, and knowing that it could have been even worse is even more troubling! The plug patch kit I keep in the car isn't always going to work. I have a second set of wheels but wonder if it would truly work ok in a pinch (235 on 8.5" wheels vs 245s on 9" ride wheels)...
I had one flat and Tesla says they had no spare wheels available. I was towed a short distance to the service center. Luckily the service center had a spare for me.
I keep tossing around the idea of dividing up the stock wheels for spare tires, but I wouldn't bring a share with me most of the time. Not a bad option to have one at home though.
The OP still never says why he was "towed to a local tire shop" instead of just getting a spare tire. I have been told that, sometimes, the towing agencies that tesla contracts with may be "out" of spare tires, but they normally have them. Like I said, unfortunately, I have had to use tesla roadside twice last year, for flat tires that I could not drive on, and I simply used the app to get roadside assistance.
One time I had the tow driver take the flat tire to my local service center. You sign an agreement that you will go "deal with" your flat at the local service center, they put a spare 19 inch tire on your car (regardless of what size tires you have on, they put a 19 inch tire on the car) and away you go.
The second time I had the tow driver put the damaged car in the trunk, and dealt with it myself by taking it to the service center in the next couple of days.
Not being used to different size tires on the car, I challenged roadside assistance when they said "our engineers have stated that we can put 19inch tires on any of the model 3s we sell. I asked for that to be sent to me in email, from their tesla account as I wanted some proof that someone told me that (being used to being told that you must have same size tires on an AWD car). They sent it to me, in an email, stating exactly what they told me on the phone.
the only issue is, the spare doesnt have a TPMS so complains about tire pressure, and they mark the loaner tire up with paint, stickers etc, because they have had a problem with people just trying to "keep" them rather than dealing with their spare.
Anyway, if the towdriver / company didnt have a spare thats one thing, but if someone just says "tow me to discount tire, ill buy a tire there" the tow driver is not going to argue with them, and then its the owners fault, not the tow drivers if they cant find a tire.