Any known cases of lucky collisions resulting in security breaches?
There have been similar "security breaches" where someone did indeed gather a lot of public RSA keys and successfully factored some of them using Euclid's algorithm.

I believe the weakness was that a lot of implementations didn't use "random enough" RSA key generators (by using a cryptographically secure random number generator) and this resulted in some common factors of said public keys.

Here's the paper with more information if you're interested.