A singularity is quite literally the name of the impossible: it’s the point within a black hole that quantum physics breaks down. So you’re correct in that event horizons contradict them. EVERYTHING contradicts them, yet there they are.

And so yes, when Hawking was theorizing black hole decay, he was specifically trying to figure out what a singularity was. He collaborated on different theories just trying to understand singularities, whether black holes had them or not, and how they might be formed. They basically confirmed that, “yep, some spooky shit happens in there we don’t have the tools to understand yet.”