If I remember correctly, when I called around to the usual suspects at the time (America’s Tire, Costco, etc), they all declined to do Teslas. Maybe their policies have changed in the meantime, but I’ve seen other posters say that their local America’s Tire, Costco, etc also won’t do Teslas either, while other locations will.
I’m going to disagree with you on the cheaper, faster, better claim also. My experience with my Volt was the dealer (!) often turned out to be the better option even on simple things like tires. I ordered new tires online, but it took 3 return visits before the tire shop figured out how to reset the TPMS. Totally unnecessary since all they had to do was mark which wheel came off which corner and put it back on the same corner. After the wasted time of 4 visits, the Chevy dealer told me they would’ve mounted the new tires for less even though I didn’t buy them there. And the Volt TPMS reset was no mystery to them. Where was the harm in asking Tesla instead of assuming like I had done with Chevy?
When my 3 needed its first tire rotation, I called Tesla to ask how much they charged. They offered a free rotation at home or work without me having to be there. I never would’ve known if I hadn’t asked. My coworkers were so impressed seeing mobile service that Tesla made a couple more sales.