There is a solution to this problem, but is only a temp fix. Go ahead and tap the button to continue and and it will take you to the recovery setup screen, then just tap the back button without inputting any information. Doing this will make Google Play think you've completed the recovery process, at least until it checks again, which seems to be every time they update the Google Play app[or 30 days, haven't quite figured out which].
I agree with you that it is a serious annoyance, but then again, if you've used the Play store on a phone or purchased anything then Google already has the information they're requesting and they simply want to encourage you to setup an account safeguard, which isn't unwise in some cases. They can NOT, however, force you to use anything you don't wish to as there are laws in many places that strictly prohibit them from forcing the public to use something they don't wish to. If they did, you'd have the full right to civil claim for damages, depending on where you live.
It's an annoyance for sure, but a minor one if you know how to work around it.