Same applies to petrol cars, many don't replace the catalytic converter they just remove it. Don't want to get in a argumentation which one is "cleaner" afterwards, both are equally stupid.
It can still happen if there are other issues such as a malfunctioning ignition system though (which could allow unburnt fuel to ignite inside the catalytic converter and cause it to overheat). This is why the check engine light actually flashes when the ECU detects a multi cylinder misfire (to signal a condition which could damage the converter).
Another note: Any car since the mid 90’s will throw a check engine light if you simply remove the catalytic converter. This will cause it to fail inspection in most states of the USA, which ultimately results in the car being legally un-drivable.
There are some people who manage to remove the catalytic converter and still pass inspection by simulating the signals from the secondary o2 sensors (or directly modifying the ECU), but this is a pretty rare practice and is usually limited to the “performance enthusiast” community. You can easily smell when a car has no converters, and it’s a pretty serious fine if you get caught.
Not if the inspection station ignores the ODBII/III tests and does a direct tailpipe test. They'll get caught pretty quickly. Some stations can do that, too. They think ODB is lying, they'll sniff your exhaust.
I have replaced two of them over the past few years. If the third one ever goes, I'm completely done with the vehicle.
But this is me, a home mechanic, doing the work. For a professional with access to professional tools, it would be a different story. The parts themselves aren't too expensive. The easy one can be purchased for <$50 online and the two others are <$200 each online.
Can you get away with this long-term in most of the EU? In most (or all?) US states, you have to have some sort of historic vehicle or something to get it registered without a catalytic converter.