What exactly means "wouldn't" and "can't"; do you get any error messages?
Can you post the contents of the FlashInstall.log file in C:\Windows\system32\Macromed\Flash?
Pat, thank you. Done.
We spent some time last week digging into this with a few people, and from what I've seen, the issue is that you need to pick up the latest Certificate Authority root certificates.
CAs are companies that validate that software publishers and the software that they publish is legitimate. They mark code as approved by issuing a cryptographic certificate to the companies they approve. The companies then use that certificate to sign their code. Your computer checks to see that the code was signed by us and was not modified in transit, and also derives the root certificate authority that issued the key, and validates that our certificate was signed by a legitimate CA (who would typically validate that we're an actual company and that the employees requesting the certificate are authorized to do so).
The root CA certificates change every few years, and you need to pick up the most current set. Typically you would get these through Windows Update, but it seems like a number of people don't have them. The following guide tells you how to pick up the latest copy for your particular OS version.
I'd really appreciate feedback on whether or not applying these updates allows the other installer that you downloaded (aih_<whatever>_install_flash_player.exe) to work correctly.