If you passed your car test before 2001I think you can ride a 50cc on that licence but after that you need a cbt before you can ride anything. This restricts you to 125cc or 11kw. So the old 2 smokes we all love if are full power can't legally be ridden on a cbt as surprisingly or not to some of us they can be up to 33bhp. Yes that's from a 125!
I have been lucky enough to own quite a few bikes in my life so far. One of them been an Italian import ns 125r that was full power 33bhp and surprised quite a few people. Other 125's barring the odd aprilia rs 125 and others like that had no chance and even some bigger 4 cylinder 600 even 750's had no chance once I got on the twisties. There is no better sound than a screaming 2 banger. The noise they make just begs you to go faster. If I was the op of this thread I would have the kh 125 that was mentioned just because it's a 2 stroke and probably the only old 2 stroke you will get for less than a grand. 125 2 strokes go for so much now because people on a cbt can ride them legally and they are so good and you won't get anything better these days imho.
The Chinese bikes you see a lot of these days aren't so bad tbh. I have owned quite a few and tbh they are just like any other bike. Ok some of the nuts and bolts shake themselves loose but check them or replace them for better ones and maintain it properly and it will last much longer. Most horror stories come from 16-17 year old riders that don't maintain or check things and end up with a knackered bike or worse. They are like any other machine, they need maintenance. If you look after it,it will look after you.