Unfortunately, it sounds like you downloaded a fake/malicious Flash Player installer file. Nothing you describe would happen if an official Flash Player installer from adobe.com (or macromedia.com) was used to install Flash Player. From what site did you download the installer you used?
It only did this on Google Chrome my default browser. Safari and Firefox still had the proper home page.
Chrome uses the PPAPI plugin, Safari and Firefox use the NPAPI plugin. They are not interchangeable.
You can try using the Flash Player uninstaller, posted at Uninstall Flash Player for Mac OS to uninstall Flash Player, assuming the official uninstaller will remove whatever malicious files were installed.
As for cleaning up your system, I'd recommend running a complete scan for viruses, malware, adware, spyware, etc.
To close the loop on this, customer messaged me indicating this was (unfortunately) indeed a fake/malicious installer.
While there is a legitimate Flash Player update notification, bad actors have taken to making legitimate looking notification dialog windows that, sometimes, are difficult to distinguish from legitimate update notifications. When in doubt, close out of the update notification dialog and go directly to get.adobe.com/flashplayer to update Flash.