So on your gaming system, you are using OBS to capture? or something else?
I know these this log is from your streaming PC? is obs up to date?
turn off Syncing audio to video time just as a test. set your audio delay to "fix" sync issues.
Stream this video to get the timing just right:
https://www.youtube.com/watch?v=stBliSJ06vc
Other thoughts:
Set your GOP to your fps, i.e. 30,48,60... Set your IDR to double your fps, i.e. 60,96,120 depending on your fps. but I dont think thats the issue. <---best practice
scene buffering: play with this setting, you might not need as high as 700ms but just play with it a bit after you turn off "Syncing audio to video"
Just my thoughts.