A bad fuel injector should usually be a reason to stop driving. Depending on the type of damage it would likely damage the piston and/or cylinder fairly quickly if you attempted to keep running it.

That would obviously depend on the failure mode. But there certainly are failure modes which could be quite damaging, and an ECU may have limited ability to determine what failure mode is occurring, and even if it has sensors that can indicate certain failure modes, it is not always clear if those can be trusted, as they there be additional failure modes that make sensors give misleading results.

So shutting it down certainly seems sensible.

There are situations where you must run the car, even at the risk of damage, because waiting for help is dangerous to the occupants.

Cars have a “limp-home” mode which they enter if sensors show odd yet not critical errors. Usually it restricts the acceleration and top speed to 30kmph or so. If it totally shut down it was likely a very serious error.

But what cash grabbing opportunity would the dealer have in that case?