My friend was able to beat his record but his recording software crashed. If he recorded a run that didn't beat his P.B. would his P.B. count as Hypixel does show your personal best even on failed runs.

You don't need to record a run. Only the top 3 requires a video proof. The rest of the leaderboard can either go for a video (optional), or you can just submit it with a screenshot showing ALL CHECKPOINTS being reached and the final time.