1st gear is pretty low. I think I only ever drop into 1st gear once already stopped.  Are you saying that it dies when shifting to down to 1st after stopping?  Or that it tends to die when braking hard at lower speeds?

Personally Ive had issues with my bike starving for fuel when braking hard. Or maybe even its flooding, not sure. About the time Im coming to a stop it will starve out (idle drops too low) and it has died from time to time, but not just slowing for corners, maybe a U-turn. Otherwise Ill carry the speed.

Do you go down to 1st gear for corners?

Its this thing, you only have one cylinder, one carburator!
In any case.. I got the same problems riding it home for the first time, almoste no idling at stops.. my bike is only 20 years old, but a low milage.
I did take a longer road home, such a ones whitout any trafic jams.. to be safe!

Soo the second ride, to my local MC-shop, for new tires, tubes, oil and oil filter.. WTF, trafic jams, I have to pull aside several times, to start the engine again, if was last summer!

This summer, it was runing as it should, exept one time, i have to lower the trothle sudenly and rise it up again, several times to make it going even. turned the petcock to reserv and aimed for nearest tank station, to refuel. I couldnt put that much petrol in my tank?

Soo Im a bit in to that of carburatuer cleaning, looking att several Youtube guys abot that, but Im not that asure, but if I can find such expert Carburatour cleaning guy, I would probably let him have a try?

And then I talk to my biker friends.. they never empty there tank or carburator in winter, its this its runs like nuts in the spring, then it runs better after a wail?

I cant come to any hard conclusions, but my friends do have at lest 2-4 cylinders and often fuel injektion!