Are there any real-world examples where encryption backdoors have been successfully used without compromising cybersecurity?
No. Adding a backdoor to cybersecurity is fundamentally introducing a vulnerability that can be exploited by an attacker.
A backdoor in your IT security is like a hidden button to bypass the lock on the impenetrable front door of your impenetrable house. Sure, it makes the police serving a warrant easier, but now there’s a button that anyone can push to bypass your door.
What you will find are instances with no apparent violations. Just like setting all the nuclear weapons to have the exact same easily remembered activation code didn’t actually lead to a nuclear exchange.
I see you understood the point made by the example.
For nuclear weapons specifically, the activation code was supposed to be a command and control measure to prevent unauthorized use. Having it both be an easily remembered code and one widely known made that whole system meaningless theatre.