I have done some tests today to look if this problem does also occur
with other distributions and kernels:
Via Technologies VL805/806 xHCI USB 3.0 Controller working or not
working depended on Linux Kernel version
Distribution Kernel works or not
Debian 11.x bullseye 5.? yes
Debian 12.1 bookworm 6.1.0-10-amd64 no
CentOS-Stream 9 5.14.0-354.el9.x86_64 yes
Manjaro 22.1.3 20230529 6.1.30-1-MANJARO yes
Fedora-Workstation 38-1.6 6.2.9-300.fc38.x86_64 yes
Ubuntu 23.04 6.2.0-20-generic yes
I also tried to install testing. But it did not install grub and could
not solve this with rescue-mode. But during testing installation the
backside USB-port mouse and keyboard also stopped working after some
time. I continued with a second mouse and a second keyboard on front
USB-ports.
Best regards
Rolf Reintjes