Replacement gear for Tidmouth Shed turntable UK

Yes that middle screw holds the platform in the center

I'm gonna upload all the pictures I took when I last opened it just for your reference. They may not be much help but it's something

     

Also there's an off white plastic cover that holds the pieces in image 6 it looks like this
 
That plastic cover causes I bit of friction to the white and orange piece shown here

That's specifically where you're getting the issue of the hold and release mech not releasing (or in other words not coming back up to hold the train) the spring. They're 4 screw holes that are responsible for holding this cover tight so maybe add some grease to the area that they rub against and also when putting the off white cover back on, play with the spring mechanism to see if it springs back easy.  You have to play with it a couple times to see if it gets stuck eventually.  If it does, then slightly loosen the screws nearest to those plastics until you get the desired result. (don't loosen them too much because then the plastic cover will interfere with something else that I tend to forget what it interfered with. Just loosen it slightly and slightly until the orange and white plastics are no longer hindered by the friction of the plastic cover)

And yes, that rotational problem sounds like a broken gear to me. You can rotate a gear in one direction and it'll rotate fine enough but then snag in the other direction it depends on where the crack formed and severity. And as I'm sure you figured out there are two gears responsible for turning the turnstile in the same direction (obviously for when the train comes in two different directions towards the turnstile) 

Just an FYI, even though you can use your hand to turn the turnstile counterclockwise or clockwise the product is meant to always turn clockwise because engines can't rotate their wheels backwards on the turnstile while being held in place at the same time. I only say this to you in case you start messing with your stopper mechanism as you rotate the turnstile to make sure everything is okay as you put it back together and you're rotating it left and your stopper wheel doesn't seem to stop the turnstile at the designated track locations that's because you're turning it the wrong direction it only goes One direction. If you know this already, then my apologies.

And of course, when it comes to that strip screw situation there are a plethora videos on how to unstrip a screw so I won't add my input on that one since they're better people out there that can aid in that