Follow up!
Engine rebuild helped, but didn't fix my moped. So here's how I solved the problem of why it wasn't running properly:
Things I tested and how, with reasoning:
We need to test: Suck Squash Bang Blow. These are the 4 things and engine needs to work. These are the 4 strokes of a 4 stroke, but a two stroke engine does suck/blow/squash in 1 stroke and bang/blow in the other (they overlap). If these are working, the engine's working (and producing power). They were all working on my bike, so one of them obviously wasn't working well enough. I also replaced the belt/rollers which were both way too far gone.
Spark - bang!New spark plug. The bike was running, so that's the best test for if you've got a spark, I replaced it anyway because it's a service item and it wasn't looking all nice and chocolate covered.
Carb - suckBike still wan't running properly, so I took the whole carb apart and cleaned the inside and outside with LOTS of carb cleaner. Checked everything for wear, I'm using the standard main jet, so I know I haven't got a fuelling problem. There's quite a lot of fuel in the intake though... It's soaking into the air filter. More on this later
I tested the enricher (auto-choke) by plugging it into a battery charger and watching it stick out. Plugged a multi-meter into the loom and tested for 12-14v AC, which was there perfectly, and checked that the plug was working. So that was working perfectly and not doing the over-fuelling. I checked the vacuum lines to the fuel petcock and fuel was being delivered properly.
The actual engine - squashThe engine uses the following things to create compression: Reed valve to stop the air going back out of the intake, piston rings; for when we get near to the top of the stroke to do the final very high (120+psi) compression and the exhaust; which is just a limited diameter pipe which doesn't let all the air out in one go. I replaced the rings and piston, removed/cleaned the barrel, checked the head for warping, cleaned the intake and exhaust channels. This helped a little because the rings were obviously not working, but did not fix my problem.
As you can see, the only thing left is:
Exhaust - blowI've had my new exhaust on for about 3 months, and I had a bit of hassle with it in the first place, but it was brand new (pattern part) and I'd had one from exactly the same brand and shop before. When I took it off the bike (just as a final thing to rule out before I either took it to a shop or bought a new engine) I blew through the exhaust as hard as I could... And it was like blowing through a tiny straw! It was very blocked!
So I got a new exhaust, blew through it... And it was completely clear, I couldn't build up pressure. Slapped the new exhaust on and the scooter is absolutely flying now. New belt and rollers (6.3g) have made getting to 45mph easy (the limit of the gearing with it's undersized variator)
Why it was broken for people who care about the physics/engines:
Here's what was happening: The compressed air and fuel in the cylinder were ignited by the spark plug, this caused the piston to move down, suck air into the crank case and opened the exhaust port where the gas should have been escaping (but in reality, it was compressing it) when we reached the bottom of the stroke, the compressed crankcase air got sucked/forced into the barrel above the piston and started to get compress. The exhaust was under such pressure, that this stopped all of the clean air going into the cylinder for compression and actually forced it back out of the intake (with some of the petrol, which is what I saw in the intake) and lowering the oxygen/fuel mix enough to stop the bike running properly, but not enough that it wouldn't drive/idle.
So, that's my adventure into 2-stroke engines, which I didn't know much about before, but am now an expert on (unmodified only!)