My knowledge of grammar isn't great, but isn't bike(-ride) working as a noun here? I can use it as a verb and say "I biked here this morning" or "it's faster if you bike across the bridge". But I've never heard "I'm going for a bike", or going for a "20 mile bike". Bike ride is what sounds natural in those sentences.
And yet you can "go for a run/jog/walk/swim". English is weird. I guess it's because bike as a noun already means the bicycle itself, while there is no object named "run/jog/swim".
And yet you can "go for a run/jog/walk/swim". English is weird. I guess it's because bike as a noun already means the bicycle itself, while there is no object named "run/jog/swim".