I have lost coins stored on USB flash drives when they broke down. You should also back up to another option like a CD. It's not only old USB flash drives that break down, it's new ones too. I bought two identical drives, and one had broken within a month. The controller chip had broken inside it, which stopped windows from recognising it, so I couldn't try using data recovery software to get my private keys back off it.
Backing up the private keys in multibit classic is fine unless you want to start using them in multibit HD. Unfortunately there is no option that I know of to import a multibit classic private key into multibit HD. Nevertheless you can import a multibit classic private key into other wallets like electrum or core.
IMO multibit HD should have an option to import multibit classic keys because people expect backwards compatibility.
You can do a bit better. First of you can add the private keys in a password protected word or excel document and then add it to a .rar again password protected document. Store it in 2-3 USB-s like you have done (chances are low they to break down in the same period) , also on CD, external HDD.
You can add this document to Google driver or One drive (not sure if .RAR password protected documents are allowed but they should be normally).
As a last advice before judging that the inside chip of your USB has been broken try the last resort Ultimate Hiren Boot CD to check if the USB is really broke or it's just windows.