NVidia 510 broken after automatic kernel update and updated kernel not supported


I am unable to enter my Ubuntu 20.04 after (I think) a kernel update (I get a black screen and have to use a tty terminal). I was on 5.16.0-11.1-liquorix-amd64 kernel since the my drivers broke a few weeks back (this thread: 510.53-RTX3070-Ubuntu 20.04 Blank Screen and dkms unsupported kernel when reinstalling). Now running

sudo apt install --reinstall linux-headers-$(uname -r)

it tells me the headers for this kernel cannot be downloaded (and Nvidia drivers do not work even with a full reinstall). In the updated kernel 5.16.0-16.2-liquorix-amd64 I tried installing the NVidia drivers, but it tells me that the kernel is not supported (I am installing via sudo ubuntu-drivers autoinstall)

NVidia bug report for the 5.16.0-16.2-liquorix-amd64 kernel: Ubuntu Pastebin

Is there any way to get either kernel up and running again. I am also happy to use any other kernel that you think is stable. Any help is greatly appreciated!

This looks like the headers for the new 5.16.0-16.2-liquorix-amd64 kernel are broken, please try to reinstall them.
sudo apt install --reinstall linux-headers-5.16.0-16.2-liquorix-amd64
and then check
dkms status
if the driver is installed for it.