"Cycling has become an addiction" sounds like it's covering up the actual issue(s) troubling your wife. I'll guess that "addiction" means bicycling takes up a lot of your time, but is that really the underlying issue? If you didn't ride at all for, say, a month or two, what would you do instead with the time that you normally would be out riding? Would you and your wife do stuff together? Would you take care of household chores that she now needs to do because you "don't have the time"? Would you go on a trip she's been wanting to do for years and now you can because your empty nesters? Etc.
Or is it the money you spend on bicycling? Your wife thinks it's excessive, you can't afford it, it should be spent on the kids/house, etc.?
I've been married about the same amount of time, and usually when my wife complains about something kind of general ("bicycling has become an addiction"), there's other, more specific issues at the root of the problem.