If there's no video proof of all of these runs, then how are they verified?

I think a good reason why there's no video is because, and hear me out if it isn't in the rulebook, the World Record run requires video proof.

Do you need video proof?

Since the primary moderator has changed to me, video proof is now required for all submissions.

Getting 3DS footage is hard, almost impossible to get a capture card.

You don't need a capture card. The low quality video captured by a phone camera or webcam is allowed for almost all DS/3DS games.