https://www.klipper3d.org/Installation.html
https://www.klipper3d.org/FAQ.html
Did you check which serial port your Pi zero uses? https://www.cyberciti.biz/faq/find-out-linux-serial-ports-with-setserial/
From line 12072 in your klippy.log is written: mcu ‘mcu’: Unable to open serial port: [Errno 13] could not open port /dev/ttyS0: [Errno 13] Permission denied: ‘/dev/ttyS0’
That is not ttyS5!
What did you define in your printer.cfg?
PA9 and PA10 should be ok.
Good idea.
This thread may help BIGTREETECH-GTR V1.0 menu config - #11 by phongshader
Good luck, hcet14