The Flash Player download page does not use the SpeedTao Download Manager. This must've been installed on your system at some other time. If you suspect SpeedTao is interfering with the Flash Player installation, and don't require SpeedTao on your system, you can remove it using a utility such as AppCleaner, or delete 'all' SpeedTao files (/Applications/SpeedTao.app, ~/Library/Preferences/net.speedtao.mas.SpeedTao.plist, ~/Library/Application Support/SpeedTao). Removing just SpeedTao.app from /Applications doesn't completely remove it and Safari will still list it in the Plug-ins (I verified this before posting my response). If you require, or don't want to remove, SpeedTao, you can try using the full installer, posted at the bottom of this page, Installation problems | Flash Player | Mac.
Thank you for your quick response, Maria. Actually, trying to download the full installer gives the same behaviour as I explained for trying to download the uninstall .dmg: If I click on it, it only refreshes the page and nothing else happens. The strangest thing is that I could not find any of the SpeedTao files you mention on my mac, so I could not delete anything... Even though Safari shows it in Preferences, at manage website use of plugins... Quite a mystery.
So I have now tackled the problem from the Apple-side. I checked on the Apple site what they said about SpeedTao and saw that I could download it as a standalone application, where you can drop download links. So that is what I did: I downloaded and installed SpeedTao and dropped the adobe link for the full installer in it, and that worked! I have flash player installed now and it works like a charm... Not sure yet if the download center 3-step routine will work too now, but I will check that when I need to download something (don't want to possibly mess up something that is working fine now ).