apt's kernel update process is not broken, you've just somewhat borked your system either by removing the kernel meta package or messing with repository sources/priorities.
Make sure "linux-image-amd64" package is installed for Debian (and whatever equivalent package name for Ubuntu) and "apt upgrade" will install new kernel packages just fine.
(And this is actually not even needed for most kernel updates in Debian because most of the time package name stays same and updated in place, package name is only changed if ABI is changed and requires recompilation of DKMS modules)
Make sure "linux-image-amd64" package is installed for Debian (and whatever equivalent package name for Ubuntu) and "apt upgrade" will install new kernel packages just fine.
(And this is actually not even needed for most kernel updates in Debian because most of the time package name stays same and updated in place, package name is only changed if ABI is changed and requires recompilation of DKMS modules)