Hey, I had the same thing just happen... looked like it was something to do with that riot client... i was able to fix it with these steps (hope this helps):
Solved this... uninstalled the app from my computer... went and found any remaining files that were league related... found a couple folders in a shared space that were Riot Client and Riot Games... deleted any folders / files i found that looked like they related to Riot or League of Legends... Re-installed and it worked.
Also i ended up updating my OS to Catalina 10.15.1 and was still able to do the install.
but it does look like it's using the old login page not the new riot one anymore