What's wrong with my scooter?

So

I have a small problem with my scooter. First of all: it's a 2001 Speedfight 2. 50cc, 45 km/h. The belt and converter were replaced a few months ago because the belt broke. The previous owner said that his previous owner had installed a new carburetor, which is clearly visible. My previous owner said the scooter sat in the garage for a year and was occasionally started. Then he got it running again and sold it (in this case, to me). The scooter ran well, and everything was great. Until not too long ago.

Now to my problem: If I ride the scooter for a few minutes, about 3-4 minutes, it's warmed up. The problem, however, is that when it starts moving from a standstill, it barely moves once it's warmed up. I give it full throttle, it starts moving, and it sputters. Then it takes about 5-10 seconds to get up to speed. At around 10 km/h, the scooter pulls again, but not with full power. After that, it struggles to reach 45 km/h.

When it's warm, the idling noise is different, and it revs much lower. I have to turn the idle screw all the way in to get it to stay running.

Does anyone know what all this is about?

One more thing before I forget: I know it's pulling in some air leaks, but it's been doing this for a while. I just don't know where, and it didn't cause any problems before. Could that still be the cause?

I don't ride the scooter often, usually 2-3 times a week for 5-10 minutes.