Do I have to record the game and splits toghether, or can i submit a split-less run as proof?

You don't have to submit those ^^ It isn't required to have splits in your video

You can submit runs without splits, but showing them is appreciated.

My laptop can't handle recording the game and the splits together, so I just record the game but still do the splits and upload them to splits.io and add to the submission (or at least that's what I do currently)

Edit - I do also add a generic timer to the gameplay video in post.