This is probably going to end up being a question for Microsoft, and the method by which you originally deployed Flash Player to your client machines, and the method you used to upgrade the machines probably bears on what the actual problem is.
There are three variants of Flash Player on Windows:
- The ActiveX variant for Internet Explorer and Edge, which in the context of Windows 10 in a built-in component of the operating system and distributed exclusively by Microsoft
- The NPAPI variant for Firefox and other Netscape-based browsers. This is a separate piece of software, stored in a system location that's shared by all browsers of this type
- The PPAPI variant for Chromium-based browsers. Google Chrome has it's own built-in Flash Player, but you can also theoretically download a PPAPI plugin that lives in the system location like the NPAPI player. (e.g modern Chromium-based versions of Opera)
Only Microsoft can write to the locations in the filesystem where the ActiveX Flash Player lives on Windows 8 and higher. That's why Flash updates come through Windows Update for that target. If you need to delete files from protected system locations, they're probably going to have to help you figure out how to do that.
The uninstaller *should* work just fine for the NPAPI and PPAPI variants of Flash Player, when run as a user with Administrator rights. I *have* seen people use third-party tools to yank files out of those directories, but it can definitely leave the system in a broken state. (Hacky tricks like removing the last Flash Player update and IE update through Add/Remove programs, then reinstalling through Windows Update will usually, but not always get people that have done that into a good state again, or a system refresh in a pinch.) System File Checker is also an option, but I don't have great luck with it for this class of issue.
There are also ways to manage Flash Player deployments in an enterprise context. It sounds like you're small enough that you're probably not doing that, but if you *are* using something like SCUP/SCPD to do deployments, then you may want to look at the system administration guide for ideas.
https://www.adobe.com/content/dam/acom/en/devnet/flashplayer/articles/flash_player_admin_guide/pdf/latest/flash_player_32_0_admin_guide.pdf