The download that "locks up" is a "stub" installer. It's only 1mb and it downloads the rest.
The one you need is here: Flash Player 126.96.36.199 Plug-in FULL Installer (Mac OS X)
Download that by right clicking (or Ctrl+clicking) and selecting "Download Linked File as".
Save it and quit Safari.
Double click it and it will "mount" the DMG and open a window.
DO NOT run the installer from there!
RIght click (or Ctrl+click) the installer and "Show Package Contents".
In Contents/Resources, double click the Adobe Flash Player.pkg
A "COCOA" installer will open. Follow the instructions to install Flash Player.