I do that. This was never a problem, as the ANC ones I used don't cancel every sound the same way.

For example, I can go into datacenter and it will cancel all the datacenter noise(aside for when air blows directly into mic, it overdrives it) but I can still hear what other person is saying.

Also I used them to generally listen to podcast so there was no wall of music to go thru, so sirens and such were easily discernable

You do you but as a cyclist you are super vulnerable to all manner of things and I'd never want to give up that kind of awareness.

If you listen carefully you can usually hear a cyclist behind you who may want to pass or is passing you, and having headphones probably makes that a lot harder

>I do that. This was never a problem

The most problematic people in traffic are never aware that they are the problem.