you can't. Only on pc you can

you need a capture card and then use it on pc its the only way

If you have a capture card you can use LiveSplit but you'll have to split manually and loads will be counted in the times

Is it mandatory to use stopwatch in outlast speedrun? Any%

It's not compulsory, I make my runs without a stopwatch

Yeah you don't need one, we retime the runs anyway. It's more convenient for us mods though if you do have one.