Your GPU is maxed out. OBS uses your GPU to render frames and composite the scene. If your GPU is maxed out, OBS has no GPU resources to composite the scene and render frames. To solve this problem, cap your frame rate.
16:13:04.921:┗output_frame: min=0.267 ms, median=0.874 ms, max=29.352 ms, 99th percentile=2.557 ms
16:13:04.921: ┣gs_context(video->graphics): min=0.12 ms, median=0.668 ms, max=28.944 ms, 99th percentile=2.206 ms
16:13:04.921: ┃ ┣render_video: min=0.027 ms, median=0.182 ms, max=10.044 ms, 99th percentile=0.945 ms
Output frames are around 1 ms. Should be at less than 3% GPU usage if he's rendering at 30fps. He has a GTX 970. There should be no problems handling 1080p canvas.
Renderer lag also will not report encoder overload errors.