Do you think it's even theoretically possible that an air traffic controller will say "looks like you might be having trouble with your plane... we're not going to allow you to land here. Go elsewhere"

It is much more likely that Air India dispatch wanted the airplane repaired at a base where they have maintenance.

I also suspect it was better for the crew to end their last airline flight as crew in their home country...

A few years ago, was on a Delta flight out of Detroit to Boston. Over halfway to Boston, we turned back as the airplane had a maintenance issue that Delta wanted to deal with at DTW rather than BOS. (They do have a permanent presence at BOS, but a larger one at DTW.)

It would be the opposite, every major airport will be open for a plane in trouble.

No

No. That's really not how anything works.

No.

1. You would not want to ascend to flight level 360 (particularly with a damaged plane).

2. You would not want to leave the vicinity of an airport (particularly with a damaged plane).

Easier, but more dangerous; surely not protocol.

You would prefer to stay near an airport, rather than far away over the sea.