Since the mid 1990's, each carb is set up by the manufacturer to give a certain emission level on a specific engine. That is why it has an emission label on it - engine family etc.. The same model carb may have many applications with varying jetting. Fortunately a gasoline engine will run in a range of mixtures and not just at one specific mixture. Thus, a generic carb, with generic jetting may substitute on an engine for the original. The bad news is that unless you know the seller of a specific carb, you do not know its history. It could have been tried before and worked bad so someone returned it. Then years later it gets sold as a NOS carb. Likewise, if the carb you bought prior does not work right, you might return it, say to Amazon, and then it becomes someone else's problem who may experience the same issue you did. There is NO substitute for understanding troubleshooting on engines. ALL OF US have struggled with engines that do not run right. Sometimes replacing the carb with an original works perfectly. Sometimes it does not, but another identical to it works fine. I would look for vacuum leaks around the gaskets or intake manifold, head gasket etc.. The difference between a low emission engine and another may be as simple as different jetting, or perhaps the breather system, or maybe adjustments and defeat mechanisms to keep consumers from being able to retune the engine.
Go to James Condon's youtube channel and watch a bunch of his videos. You'll see a pattern emerge when troubleshooting carbs.
been watching cordon's channel for a long while, been there and done that many times myself, problem here is using my skill against getting information, when I search B&S carb models for this engine all I get are part numbers, not actual jet sizing, then by chance if I am lucky I will find posts of all Brigss carbs and jetting, then have to convert that sizing to my wire drill sets and it doesn't equate, see my isssue here, when this mower does start coughing and missing it is a definate carb issue I wasn't given it with the OEM carb it came with an ebay special and the air mixture screw was siliconed over. When I finally got mad enough to get a new carb yes it went from "Fits Briggs 14-17hp" on the listing, which to me being a retired wrench, sent up immediate alarms, After install the smell and blk smoke, wet plug when pulled indicated one thing, way tooooo much gas, now, is it the emulsion tube being sized for a 17hp, orrrrr the jetting sized for a 17hp, or according to TB the MAJOR warning on their manual I found that states "The 15.5hp is a low emission engine designed for lower emission and as such requires oem specific replacement parts" Hmmmm makes one think, did they actually design a fuel and valving system that was of lower emissions?? specific jetting and emulsion tube just for the valving design on this 15.5 which seems a "bastartd engine size and design to start????? Hmmmmm makes one think, In my time I have gone from Deer, Scotts, TB, Kohler, B&S, from 12hp murrays to twin 22 HP deers to the 33 HP gen set we have for hurricanes, all the way to a 25kw gen. with a water cooled mitsubishi in line 4 cyl. for a drive mtr. So you might understand, "just slapping a generic carb on this TB to me personally is an issue IF , and I say IF I cannot verify JETTING is the proper size for THIS specimen of small trailer tug I use it for. IS that a better summation of where I am right now??? I got a "Specific listed carb delivered last night, with new plug, and vacuum tube from fuel tank to carb with new a/f, after I drain all gas, change the coil, I will install new carb and see if the jetting allows for this engine, I have checked valve lash, checked PCV system, (yes it has one) no white smoke whatsoever, blk and major coughing when spitting, that alone says either valves, orr excessive fuel issue, I am leaning toward fuel as when I do get it to run at mid to high rpm's man that motor will run. Let me know more thoughts on specifics of these 15.5 carbs if you can and as always THANK YOU for replying to my posts, Even at my age, I still rely on opinions of others before I decide my fate...........red PS yesterday when it started running rough, I noticed the vac line from gas tank to carb had fallen off, replaced it and it took off running great then?? out of no where almost like it loaded up, blk smoke and that "too rich smell one gets" then back to coughing, I will see today, wish me luck!!