I had the same issue with Costco and another tire shop refusing to patch a nail in my 2020 Model Y tire.
I tried a Discount Tire - they fixed it within a few hours for free, with a smile. They only asked I consider purchasing from them when I'm in the market for new tires.
I called Walmart and they said they could do it, we shall see...
That's not to say I haven't driven a plugged tire until it's bald. And that's not to say that I wouldn't tell my sister to go ahead and get that pricey tire patched. But if you think that repaired tire is now as good as new, you are wrong. Would I do a track day on a repaired tire? Would I whip my Tesla up to ludicrous speed on a patched tire? Would I ride a plugged motorcycle tire any further than I have to? Fuck no. If I can afford a track day, I can afford a new tire.
John Dunlop didn't invent the pneumatic tire, but he put it into production. Dunlop's son was getting headaches riding his tricycle, so John D. softened the ride (as pointed out by sibling comment).
And so using air gives us three things that apparently haven't been found with any other material in combination: minimal weight, cushioning, and it can support the weight of the vehicle. If it needs replacing, one need only find an air compressor.
And I agree if it's not in the sidewall or something it would definitely at least get you by. until you get a cheap tire.
Patches when done correctly are more than safe enough for 99%+ of drivers.
https://www.amazon.com/ITW-GLOBAL-BRANDS-1034-HANDLE/dp/B00P...
And the few times I've aided others with installing a plug, attempting it without removing the wheel first just made it more difficult, sweaty and dirty, than had we just taken the wheel off for better access.
Sometimes you do get lucky on the fronts though, if the puncture is near the outside edge. Especially if your vehicle has a lot of caster presenting more of the wear surface at full lock.
But either way, we're deep in the realm of "sounds too much like manual labor to me" for someone drawing the line at attaching a can of green slime to the valve stem and pressing down the button. Wheel on or off, it ain't happening.
I definitely would try to avoid going to the service center for it. Discount Tire tends to be better.
I'm already having a load of trouble this morning getting a flat fixed because it has a slow leak and the tires have the acoustic foam inside them that prevents using green slime. Tesla wants $81 to fix it- if I drive an hour and a half to a service center.
I love the gas savings but I don't want to drive a John Deere that can't be worked on when things go wrong.