Many, many years ago, I operated a large truck stop in Colorado after having worked at service stations working my way through the university.
I have never heard of referring to a person who performs the relatively mundane job of patching a tire (tyre) a "mechanic." A tire, after all, has no mechanism, i.e. moving parts, whatsoever. We had no particular term for the people fixing tires, changing oil, etc.
On the other hand, one large, regional tire provider in this area refers to the those who mount tires, etc. as "technicians."
As to where to have a tire fixed, it varies. Years ago, tires were fixed at corner service stations in the U.S. where the attendents might or might not do a decent job of fixing them. In today's world, the old "service station" is disappearing and gas is dispensed from food stores, etc. Tire specialty shops abound. I probably would call such a place a "tire shop."
You can also get it repaired at a "tire store". In the tire stores they refer to those mechanics (with a limited skill set) as "tire technicians". They can handle all things related to tires, but unlike a conventional automobile mechanic, little else.