I recently acquired an FA50 with the identical problem. Would run for a few seconds and then quit. Once in a while after working with it for a long time, it would finally start and run great! Not a good situation at all. It had to be a fuel delivery or carburation issue.
I first verified there were no air leaks. I had to make a new carb bowl gasket and carb to manifold gasket. I also had to flatten the manifold side of the carb, which had been bent somehow. I used sheets of sandpaper on a granite stone to lap it flat. I took the carb apart at least 6 times and cleaned every orifice with carb cleaner and fine wire. It still would not run. Turns out the problem was not with the carburetor, but with the fuel "shutoff" valve located above the engine on the frame.
There is a diaphragm inside that is opened with vacuum from a line leading to the opposite side of the carburetor. This is a "safety" device so that if you forget to shut off the tap -and- the float needle fails somehow, fuel will not drip out onto your garage floor. Nice idea, but like most lawyer inspired mechanisms, it is not intended to make a better product, but rather to provide a shield from lawsuits. If the diaphragm degrades or gets a small pinhole in it, it will not open with the minimal vacuum present when you kickstart the bike, fuel will not flow, and the bike will not start.
The fix that I opted for was to simply disconnect and plug the vacuum line, and leave the fuel tap in the PR or prime position. I guess you may also be able to use the normal position as a manual shut off, but I won't bother unless I notice fuel leaking from the overflow under the carb which so far has not been a problem. I also bent the float arm slightly to lower the float level , which will help seat the float needle before the fuel has a chance to exit via the overflow tube.
Worked like a charm for me. Starts first kick warm or cold and runs like crazy. Of course any mod you do to a factory system is your choice. YMMV.
Have you checked the float valve in the carburetor bowl?
amazon carries a new carb replacement for 20$,this is a quick fix