Many people don’t countersteer. If they do, they are unaware of it. For example, you can ride a bike with no hands on the bars at all and steer only by weight transfer.
There has been many good wrecks that go something like this. A guy on a bike approaches a corner and realizes he is going faster than he is comfortable with. Panic begins. The guy starts wrenching the bars in the direction of the turn and actually skids off the road turning (a little bit) completely the wrong direction.
For example, he tries left lean with left steering input and actually turns somewhat to the right.
I know a guy here in dallas (used to work with my significant other) who died doing pretty much that trick. He just got his new bike. Appearently he scared himself on an elevated highway ramp. Actually turned the bike into the guardrail. He went over the railing and into oncoming traffic below and was then run over as well.
The investigation revealed that his speed was probably well under what the bike could have negotiated the corner at. The tire tracks indicate he turned the wrong way. We’ll never know what was really going through his mind at the time, but… It is a good guess that he tried to turn the bars in the direction of the turn and that doomed him.