I was wondering... Since i'm not interested in actually recording my gameplay, is there any way i'd be able to show proof of me actually finishing a game with the time a claim? If so, how? Thanks in advance.

If your game has an in-game time, you can post a picture of the IGT at the end of your run. It's, for most games, pretty reliable.

If not, you can still post your splits, but depending on the game (if it's popular or not) you might want to switch to video recording if you start to grow in the leaderboards otherwise your submissions might be refused.

allot of rule dependent categories for certain games/runs require video or else it risks being denied, also proofing with splits and giving comments on the run you did also help.

Most times a video is required to verify that runs were not cheated and its not a edited picture or something like that. Especially applies to glitch-less runs as an example.

As Gyoo said, maybe look at doing some recordings because it will be hard to submit anything without the risk of being rejected.