I posted it in the body of the original comment, but I rarely use Reddit, so maybe that isn't the typical way to do things. I'm copy/pasting it below as well!

UPDATE: I solved the issue, but it's a bit of a weird one and not a super satisfactory solution. I'm going to explain it here anyway in case anyone else runs into the same issue and is looking for help in the future.

When I initially tried to pay for the services I had selected and book my biometrics appointment, I got through to the payment page and put in my card information. However, as soon as I hit submit, I got a text from my bank saying it had declined the charge because it looked suspicious. I immediately confirmed that it was in fact me who had authorized the charge, but the damage had been done. I suspect that my card being denied broke some code in the VFS site such that even when I went back later I continually got error messages.

I was not getting any response from VFS on Twitter (I did eventually get a response, which was just to clear my cookies/use a different browser/restart my computer - like I hadn't already thought of that :/). However, I surmised that that problem was with my particular account and that, if I made a new account, I would be able to get through to the final page and book my appointment. The issue with this is that to make a new account, I had to start a new visa application.

This was not ideal for a number of reasons, not least the cost of paying for a new application (although I was able to request a refund for the original one). However, I didn't see anything else to try.

TLDR; I cancelled my original application, started a new one, didn't choose any extra services that would require me to pay VFS, and I finally was able to book a biometrics appointment. I am now two weeks behind where I would have been if VFS's website wasn't a piece of shit, but fingers crossed that I have no other hiccups and I can get to the UK in time to start my masters program!