The proprietary Nvidia driver has kernel modules that are specific to a single version of the Linux kernel. With pre-built packages that’s typically whatever the standard kernel is for your distro. If that kernel isn’t booted then you’ll have no graphics driver.
This is solved by DKMS, which will build those kernel modules for every kernel you have installed. You’ll need the kernel headers for the kernel you want to build for, as well as the nvidia-kernel-dkms package which the wiki you linked only offhandedly mentions. Whenever the kernel or driver updates it should build the required modules.