@AshCAs stated, I had already contacted support and they could not find the source of the problem and directed me to GoToRescue which turned out to be rather useless.
So here's the thing: the machine that I have been running g2a on until last month is a Windows XP machine. Yes, really. So I respect that at some point you might release a new version that would no longer support that OS, but it was in fact the old version that I had used before that gave me the cripled login screen.
So I started a Windows 7 image which I'd rather not do because this requires more resources, but this failed to run GoTo Opener with the following message:
The timestamp signature and/or certificate could not be verified or is malformed.
The Windows 7 image was fully patched all the way to where Windows 7 went into transition support - the image dates back to November 2017 and I think the last update available at that time was from June that same year. Reading Assaf's message I checked whether it could be possible that updates were still being released and indeed there were some updates, including certificates, but one required dependency (KB4490628) not being offered and requiring manual download an installation. This did not change the behaviour of GoTo Opener.
In the end I copied the 1544 binaries from the original (XP) support machine over to the Windows 7 machine, started it, was able to login and received the update to 1673 which installed and started without issues. So at least now I can start and use g2a.
But, that's not the end of it. I have this unattended client acting as a dial-in machine. Doesn't do anything, shouldn't do anything, just allow me start an RDP session with another machine on the network. You probably already guessed this is also a virtual XP machine and when I logon to it using the virtualization agent console it instantly starts updating the g2a client which then fails with a "file authentication failed" on uninshlp.dll and then uninstalls the whole product instead of falling back to the original installation. Obviously I can revert the snapshot, but this is really bad.
As I mentioned earlier, the login screen on the Windows XP machine showed the missing image icon. I found that if I point Internet Explorer (8) to https://get.gotoassist.com it tells me it cannot display the page, even after manually adding the corresponding root certificate. When I however use Firefox from that same machine, I have no issues displaying the page. When I registered to this community I noticed a message stating that LogMeIn requires at least Internet Explorer 11 to be installed and my guess is that you have in fact just recently reconfigured your web servers to actively bounce requests containing user agent strings that are sent by older versions of Internet Explorer. Why? For God sake, why? This is so pointless...
Oops, I realize I never replied as to what I eventually found as my solution.
I applied all patches that were pending and suddenly GTA was able to function. I have no idea which patch it was, but it seems like GTA is unable to run on a fresh install of Windows 7 and needs to be fully patched before you can run it.
For a fresh install of Windows 7 you need to install Service Pack 3 before you will be able to run GoToAssist. This is because we no longer support legacy versions of TLS. This is also why Windows XP will not work with GoToAssist.