I fired up TimeSplitters 2 for the first time in like two decades and immediately went 
at the aiming controls
Firing from the hip, there's no crosshair on the screen, which would be fine, but instead of a zoomed-in aiming mode or iron sights, aiming your gun roots you in place and gives you this light gun game-like pointer crosshair. Usually, aiming in an FPS game decreases your sensitivity to make it easier to make finer adjustments but this is the opposite. The crosshair is essentially a super sensitive mouse cursor that you control with a stubby little thumbstick, and it feels terrible to use.
Halo had been out for almost a year by the time TimeSplitters 2 came out so why didn't they just crib Bungie's homework 
I quickly went back and checked TimeSplitters: Future Perfect and sure enough, it just had a regular Halo-style crosshair as the default, called "fixed crosshair" in the options, with the old control style called "moving crosshair." I guess by 2005 Halo had established its control scheme as the de facto standard for console FPS controls
The trick is to not use the precise aim button ever, except sniping and stealth. You can do everything else without it and you need to be mobile, use the strafe buttons.
Annoyingly, the first level requires precisely those two things. I wanted to play the Notre Dame level where you shoot zombies with a double barreled shotgun but that's level three