A lot can be hapnin with a machine you just bought: old fuel, dirty carb, overheating. Was it running very good and performance deteriorated before it quit? Or did it quit when you hit the throttle as in choked?
What KeiHin number is stamped on the carb? See the stamp on this carb?
How well did you clean the carb? Did you inspect the choke valve (my bet is it’s worn and needs replaced)? Here is a pic of an 85 200sx QA02A carb I bought off fleabay and cleaned just for practice. Seller listed it as a junk trx350 carb. Yours will look similar except the 86 doesn’t have a removable emulsion tube-I think QA08A?
Only buy OEM parts these 200sx quads all use different carbs, so stick with your model carb and a Shindy rebuild kit to ensure best reliability and performance. Spray the whole thing down with Deep Creep, stick it in the freezer to help oil penetrate the screw threads. Be very careful removing screws esp the pilot jet if you break it off in there you’ll need a machinist to remove it. Don’t throw away any parts; there’s people on fleabay who will buy junk carbs for parts.