do i? Or can i just edit the video so the duration of it is the time that started and finished the game run?
Some games do require you to have a timer on screen, you'll have to check the rules for whatever game(s) you're planning to run. But generally speaking, no, you don't strictly need one. I'd personally recommend it you're able to do so, but it's up to you.
That particular game does not require you to have a timer. I'd still very much recommend using one as it's a helpful tool but if you'd rather not there's no problem with that.
If you're wondering which software everyone uses for the timer, it's called LiveSplit. It's very good.
Technically you don't ever need a timer for a run. Leaderboards which require a timer don't make sense to me. As long as the video is not missing any footage you can calculate the run time in an exact way and without much effort. A timer also does not add a tiny bit to legitimacy since you can just prerecord some stuff and always add the timer to the video afterwards.
YouTube's player is kind of weird when it comes to rounding the video length, so I'd be careful using the videolength as proof for that. Depending on what software you are using to edit adding a little timer to the video already helps with that a lot. I only know Sony Vegas sadly, but that has a timer effect that you can attach either to the full video of the run or you add a blank single color (with 100% transparency) object to a timeline above and add the timer to that, which gives you the option of using keyframe animation to move the timer around or scale it down if it's too large for your liking.