GIve it a COCOA install:
Save the file and quit your browser.
Double click the DMG to mount it.
DO NOT run the installer.
Right click it and "Show Package Contents".
In Contents/Resources, double click the Adobe Flash Player.pkg
The COCOA installer will open.
Follow the steps to install it.
I followed your instructions, and the install still fails. The error message is: Adobe Flash Player can't be installed on this disk. This system does not meet the minimum requirements for installation. Mac OS X 10.6 or higher is required.
As noted earlier, I am running Mountain Lion (10.8.5).
Any thoughts? For now, will go back to an old Flash installer and see if I can get it work.
I did a "safe boot" and got the installer to work. This a trick I learned from the System 9 days. Never thought I would need it again. I have no idea of what was interfering with the install.
P.S. BTW, I didn't need to use the COCOA installer.
Actually, I made a mistake. I ended up installing flash on my Snow Leopard drive, not my Mountain Lion drive.
What I've ultimately done is taken my Mountain Lion back up drive off the shelf and installed Flash on it. Next, I've spent a couple hours updating the Mountain Lion back up drive. Later this evening, I will erase the Lion drive and replace it with the contents of the freshly updated Mountain Lion back up drive.
I'm thinking that there was a permissions problem on my Mountain Lion drive that was not corrected when I updated permissions.