Logs should be located in ~/.config/obs-studio/logs/ though the output you've provided does hint at what reads as a big issue for getting the UI to work: it cannot load the text for your language:
warning: Failed to load 'en-US' text for module: 'decklink-ouput-ui.so'
libDeckLinkAPI.so: cannot open shared object file: No such file or directory
I am unable to locate that file on my own system (also Debian based) but my install also doesn't make any mention of it in the logs, so I'm not sure what's going on there. A plugin perhaps?
Logs indicate that it continues onwards, then bails just after switching to the default scene, and the last line may be referring to the above error:
obs: Fatal IO error 2 (No such file or directory) on X server :0.0.
Can you provide the output of the 'w' command? Maybe X started up on :1 and you need to start OBS accordingly:
DISPLAY=:1 obs