Also a regular bike checkup should includes tires checkup to remove the debris stuck in the rubber. Those can “move” and get you flat in the future.
Would good tires still save me from these punctures?
I've seen similar bike patching mechanics on sidewalk stations in Beijing, and I feel like they would have a similar primary customer. The degree of construction and infrastructure work there means that there are always metal bits in the road. Nails, roofing scraps, welding rod ends, filler wire trimmings that get washed off construction sites in the rain and blown to the roadside by traffic are highly effective at puncturing bicycle commuters' tires. But they also make small holes that are a quick fix. Soapy water will show where the leak is, and pliers, a donor tube, and vulcanizing compound and you're in business again.
Gatorskins are harder, less supple, so they don't pick up the smaller bits of glass and metal that will work in over time. They are resistant to pinch flats by having thicker, stiffer sidewalls, so the bigger bits of metal or rock aren't so scary ( they won't be any more fun, you will feel them more ). But a nail is a nail and if you hit it head on it will go in.
https://www.continental-tires.com/products/b2c/bicycle/tires...
Thicker tubes can help, too. I use some really beefy tubes. They're way heavier but it's worth it to me. I went from a few flats a month to no flats a year, back when I commuted by bike in Austin.
Current ones are some Continental with 5mm reinforcement IIRC. I came home one time with a piece of broken glass stuck in the tire. It was just stuck, pointy end in the tire but did not puncture it, so I just removed it and laughed.
Tubeless is a great solution for preventing punctures (especially sharp thorns left on roads after trimming of hedges and bushes), but it can be a faff to set up until you know what you're doing and the tyres are more expensive. You also "need" compatible rims to run tubeless unless you fancy a ghetto setup.