I've just tried to generate & insert r8169, by first downloading GBE Ethernet LINUX driver r8169 for kernel up to 5.19 (i.e.r8169-6.031.00.tar.bz2 from https://www.realtek.com/en/component/zoo/category/rtl8110sc-l-s and extracting it to r8169-6.031.00:
john@Desktop:~/Software/Realtek/r8169-6.031.00$ sudo modprobe -r 8169.tar.bz2 & extracting it tor8169-6.031.00 and then doing:
[sudo] password for john: 
modprobe: FATAL: Module 8169 not found.
john@Desktop:~/Software/Realtek/r8169-6.031.00$ sudo make clean modules
make -C src/ clean
make[1]: Entering directory '/home/john/Software/Realtek/r8169-6.031.00/src'
make -C /lib/modules/5.19.0-43-generic/build M=/home/john/Software/Realtek/r8169-6.031.00/src clean
make[2]: Entering directory '/usr/src/linux-headers-5.19.0-43-generic'
make[2]: Leaving directory '/usr/src/linux-headers-5.19.0-43-generic'
make[1]: Leaving directory '/home/john/Software/Realtek/r8169-6.031.00/src'
make -C src/ modules
make[1]: Entering directory '/home/john/Software/Realtek/r8169-6.031.00/src'
make -C /lib/modules/5.19.0-43-generic/build M=/home/john/Software/Realtek/r8169-6.031.00/src modules
make[2]: Entering directory '/usr/src/linux-headers-5.19.0-43-generic'
warning: the compiler differs from the one used to build the kernel
  The kernel was built by: x86_64-linux-gnu-gcc (Ubuntu 11.3.0-1ubuntu1~22.04.1) 11.3.0
  You are using:           gcc (Ubuntu 11.3.0-1ubuntu1~22.04.1) 11.3.0
  CC [M]  /home/john/Software/Realtek/r8169-6.031.00/src/r8169_n.o
  LD [M]  /home/john/Software/Realtek/r8169-6.031.00/src/r8169.o
  MODPOST /home/john/Software/Realtek/r8169-6.031.00/src/Module.symvers
  CC [M]  /home/john/Software/Realtek/r8169-6.031.00/src/r8169.mod.o
  LD [M]  /home/john/Software/Realtek/r8169-6.031.00/src/r8169.ko
  BTF [M] /home/john/Software/Realtek/r8169-6.031.00/src/r8169.ko
Skipping BTF generation for /home/john/Software/Realtek/r8169-6.031.00/src/r8169.ko due to unavailability of vmlinux
make[2]: Leaving directory '/usr/src/linux-headers-5.19.0-43-generic'
make[1]: Leaving directory '/home/john/Software/Realtek/r8169-6.031.00/src'
john@Desktop:~/Software/Realtek/r8169-6.031.00$ sudo make install
make -C src/ install
make[1]: Entering directory '/home/john/Software/Realtek/r8169-6.031.00/src'
make -C /lib/modules/5.19.0-43-generic/build M=/home/john/Software/Realtek/r8169-6.031.00/src INSTALL_MOD_DIR=kernel/drivers/net/ethernet/realtek modules_install
make[2]: Entering directory '/usr/src/linux-headers-5.19.0-43-generic'
  INSTALL /lib/modules/5.19.0-43-generic/kernel/drivers/net/ethernet/realtek/r8169.ko
  SIGN    /lib/modules/5.19.0-43-generic/kernel/drivers/net/ethernet/realtek/r8169.ko
  DEPMOD  /lib/modules/5.19.0-43-generic
Warning: modules_install: missing 'System.map' file. Skipping depmod.
make[2]: Leaving directory '/usr/src/linux-headers-5.19.0-43-generic'
make[1]: Leaving directory '/home/john/Software/Realtek/r8169-6.031.00/src'
john@Desktop:~/Software/Realtek/r8169-6.031.00$ sudo depmod -a
john@Desktop:~/Software/Realtek/r8169-6.031.00$ sudo modprobe r8169
modprobe: ERROR: could not insert 'r8169': Key was rejected by service
john@Desktop:~/Software/Realtek/r8169-6.031.00$ 

I have Secure Boot enabled in the BIOS.
As I don't want to make things worse, how do I fix missing 'System.map' file and "modprobe: ERROR: could not insert 'r8169': Key was rejected by service"?
PS there is the correct System.map in /boot:
root@Desktop:/# ls boot
config-5.15.0-73-generic      initrd.img-5.19.0-43-generic           System.map-5.15.0-73-generic
config-5.19.0-43-generic      initrd.img-5.19.0-43-generic.old-dkms  System.map-5.19.0-43-generic
efi                           initrd.img.old                         vmlinuz
grub                          memtest86+.bin                         vmlinuz-5.15.0-73-generic
initrd.img                    memtest86+.elf                         vmlinuz-5.19.0-43-generic
initrd.img-5.15.0-73-generic  memtest86+_multiboot.bin               vmlinuz.old