An update, in case anyone else encounters this issue.
I found another post describing a similar issue, which traced the problem to the file /usr/lib/modprobe.d/09-nvidia-modprobe-bbswitch-G04.conf which appeared to be blacklisting the nvidia kernel module and prevented it from starting. It turns out I had the same file.
I eventually tracked down the package that owned that file – it turns out it was put there by suse-prime, which was installed at the same time as the nvidia driver packages. Since my laptop doesn’t have Optimus hybrid graphics, I didn’t need that package, so I uninstalled it and the drivers started loading just as expected on startup. (I also marked it as “taboo” in YaST to make sure it won’t get reinstalled in a future upgrade.)
I’m still getting weird stuttering and poor performance, but removing suse-prime cleared up the main issue so I thought I’d put that out there for anybody else having the same problem.