I don’t know if this was a problem with the software update, or if it was just a one-off issue with my own laptop (a Lenovo Thinkpad T430 running Fedora 43 with Gnome), but I’ve fixed it now so I’ll write the solution here in case anybody needs it. The problem was, my computer would boot into emergency mode and give me a root terminal, but nothing else. I couldn’t boot from the old kernel (6.19.6) either because it had the same problem.
In the emergency mode terminal, I did journalctl -xb and it was very helpful. There were some errors relating to memory, so I rebooted and picked “Memory Test” from the GRUB menu. But that wasn’t the problem.
The command I needed to run instead was fsck -y /dev/mapper/fedora-root. That fixed the problem, and now I can log in as normal again. All that command does is check the /dev/mapper/fedora-root filepath (which was mentioned in the journalctl output; that’s how I knew to run it there) and try to fix any problems with the file system. So it was a very basic problem. But if anyone is new and had this problem as well, I thought I’d put it here to help them, because it’s very annoying when you run updates and they break your computer.