Splits are meant to back up video evidence but are they compulsory? Are they really important? Or is it optional?
They are completely optional. They don't do anything to "back up" video evidence (if there's video you can figure out what the splits should be from that, and if there isn't then it's just meaningless numbers with no context). It's just extra data you can attach to your run if you want.
That said, there's no reason not to include your splits, and I would always recommend doing so if you're able to.
There may be some games that require splits, so always check with the mods of a game. You may see this in games with an autosplitter that removes loads from the timing of a run. Without the splits it might be harder to verify the final time of the run.
Every community policies these things differently, so it’s always a good rule of thumb to double check with the mods of the game.
Small adjustment to the above: Some games require the "LiveSplit timer" itself to be shown. Usually because of functionality (plugins) within LiveSplit that allows LiveSplit to read the respective game's memory and can therefor operate as a separate timing method. Mostly used for tracking loadless real time or tracking the game's in-game clock.
Splits themselves are completely irrelevant seeing as splits in LiveSplit or any other timer program operating similarly are purely meant for tracking the times of invididual segments.