11:02:34.826: Windows Version: 10.0 Build 17134 (revision: 165; 64-bit)
Windows is out of date. Read the link at the bottom of this post to see how to update Windows and/or deactivate game mode, which hurts OBS performance and is on by default and cannot be switched off.
11:02:35.095: OBS 21.1.0 (64bit, windows)
OBS is out of date. Download the latest directly from the site, don't use the auto-update feature, in that version it was broken.
11:15:16.866: Output 'simple_file_output': Number of lagged frames due to rendering lag/stalls: 60 (3.6%)
GPU Overload.
GPU Overload Issues
11:15:18.220: Video stopped, number of skipped frames due to encoding lag: 1095/1262 (86.8%)
Encoder overload.
General Performance and Encoding Issues
If turning off game mode alone doesn't solve the rendering lag, you can either cap your in-game framerate and/or target 720p60 or 1080p30 instead. Either of those methods would likely address both the GPU and CPU overload.
However you should probably be using the NVENC encoder instead anyway as it will remove all the CPU load for encoding purposes.