Flash player is from Adobe. System Updates are from Microsoft. They are two unrelated companies. Frankly Microsoft wishes websites would use their Silverlight product rather than flash so they could flush flash
What that means is system updates have nothing to do with Flash, Flash has it's own annoying updater. Personally I disable them both but I can be responsible enough to update my own periodically, most general users are better off leaving auto-updates on. For Flash updates I manually run
Alternative Flash Auto-Updater; you might download and give it a try just to see if it can get you current right now.
There are several things that can prevent installs: insufficient HD space; your not logged on as an administrator; your firewall may be blocking the download; your anti-virus may prevent certain programs from running (usually this last case is for potential threats, not bona fide programs).
My guess is for you that your firewall is blocking downloads or you are not logged in as an Administrator.