it has been bothering me.
A backup is something that you rely on, and that should give you peace of mind. Even though it's not strictly necessary to wipe the backup, from what you've said, it could be worthwhile if that is what is necessary to give you peace of mind.
On the other hand, having only one single backup isn't necessarily the safest thing in the world. It's better than none, but far from perfect. I usually recommend that people maintain two separate backups as a minimum, ideally created with two different programs (such as Time Machine and Carbon Copy Cloner, which is what I use). This protects against loss of data due to corruption in one backup.
Instead of wiping your current backup, perhaps you should just start some supplemental backups. Get an additional external hard drive (or two) and use some other backup tool to back up to it/them. I maintain two additional hard drives, beyond my Time Machine backups, and one of those drives is always off-site (in a bank safe deposit box). I swap those two drives periodically, so even if there's a fire or burglary, I still have all the data on the drive that's at the bank.