The launcher is actually using Internet Explorer (even if you use another browser normally) so if there's issues with IE's cache, it can make the page appear blank like you're finding.

To fix, load Internet Explorer (not Edge!) and go into the Options (The gear icon in the top right then Internet Options). On the first page in the "Browsing history" section click "Delete". If it's ticked, untick "Preserve Favourites website data" and then tick "Temporary Internet files and website files". The rest of the page you can tick/untick as you see fit (if in doubt, just untick them all). Hit delete and then wait a moment before opening the game again.