I think I just found a working solution. According to this answerhub: "Error: Did not get a response. Are you connected to the internet?" - Getting Started - Unreal Engine Forums
Carlos responded with:
“We use the windows internet options for connections and we recently found out that some programs can disable certain TLS and SSL settings. Can you open up windows Internet Options (either through control panel or running msconfig.exe->Tools->Internet Options)? In the Advanced tab, scroll down the settings and make sure that “Use TLS 1.0”, “Use TLS 1.1”, “Use TLS 1.2”, “Use SSL 2.0”, “Use SSL 3.0” are enabled?”
I opened the Internet Options windows, opened the advanced tab and scrolled down to find TLS 1.0, TLS 1.1, and TLS 1.2 were all unchecked. I checked them and hit apply, now the launcher finally loads for me, no problems.