That's interesting, as a native American English speaker approaching 40, I have never heard anyone say "go for a bike". Neither native speakers nor people whose first language is not English.
Is this common in some part of the country, or outside the US?
Yeah, from western Canada, i'd go biking, or go for a run, but I don't think I've ever said, I'm going to go for a bike. I'd say "I'm going for a bike ride".
I've never heard anyone say it in the UK either. I think the OP's reasoning is off. Just because 'bike' can be used as a verb doesn't mean that it will get used in that way in all situations.
Is this common in some part of the country, or outside the US?