This is how it works on the Jersey turnpike but the IIRC right to tow people from a particular stretch of road on a particular day of the week is bid on like any other state contract (my details may be a little off but point is you can't call whatever company you want).

Same on New York thruways. Each section is leased to a single tow company that will "cut you a deal" on the tow if you go to their shop for repairs.

This also used to be very common on the GRA, the motorway that runs around Rome (Italy). I was a victim of such a scheme once. If I remember well, the problem was finally solved due to the fact that at some point tow trucks were banned from the GRA altogether.

So what happens when your car breaks down? Do you need to push it yourself?

The tow is managed by your insurer. You get a fixed price or reduced price depending on your insurance. You get a toll free number and usually instructions not to trust random towers you didn’t call for.

My understanding is that official ACI[1] trucks, as opposed to private ones, are still allowed, if your car breaks down on the GRA your insurer will send an ACI truck

[1] https://en.wikipedia.org/wiki/Automobile_Club_d%27Italia

In some stretches of highways in India, people will throw nails on road and the only nearby repair shop will be theirs.

That's a world-wide practice, I can assure you of that.