Not sure if you've tried this, dismissed it or even if it applies to a BS carbi but I watched this video a while ago on how a carburetor works - it explains the four circuits of a carbi. What you've described here sounds like the starter circuit in the carbi not working but the main circuit working fine.
Here is the video:
and at about 3:12 it shows the 4 circuits. Maybe you just need to clean the carbi?
I'm assuming all of that's true without actually watching the video but it's from one of those engineers standpoints.
Technically you could break down the circuits into four circuits of a carb like that but people really don't.
Sometimes for clarification or educational purposes I will tell customers that there's two basic circuits on a carb and I don't mean the ones they're talking about. I mean the air and the fuel because people think sometimes that a dirty air filter or dirty Dusty conditions causes the mower not to start and run after stuff gets in the carburetor and clogs up the jets.
I explained to them that that's not the case and all the clogging internal carburetor passages has to come from what's in the gas tank or from the fuel itself congealing etc or water grease bubbles what have you.
The fuel and air never actually mix at least not until the very very end and only when the engine is running. No matter how much dirt and dust you operate the lawn mower in even if you do it without an air filter, which is bad on the engine especially the rings and cylinder walls etc even though people have gotten by with it for years, you're not going to get dirt into the carburetor bowl and clog up the jets.
But as far as what this video is relating to most people just refer to the running circuit and us old text or younger ones with lots of experience especially with older equipment will refer to the primary and secondary circuits or Jets or the high speed and low speed.
It's a little clearer when you start looking at the two cycle or similar engines with the very small carbs with two separate mixture screws.
One literally controls the gas flow at larger throttle openings and the other one controls the mixture at almost closed throttle idle.
In the old days pretty much all equipment had two mixture screws along with your minimum idle speed. Lots of people confuse idle stop screws for mixture screws.
You had the large one you could operate with your thumb on the bottom of the bowl that screwed the needle up and down and adjusted how much fuel could get sucked through the carburetor from the Venturi effect of the air going down the throat with it running then you had the smaller one up higher on the carb that controlled your idle mixture when the engine was running at idle and much slower speeds.
What they're going to call the choke or starting circuit most people just refer to as a choke or whatever it doesn't really deserve being called its own separate circuit.
Again, in the old days you had a choke that was a butterfly that closed just like putting your hand over the throat of the carb. Then they went to a primer bulb which squirts gas right into the carburetor actually by blowing air into the bowl which displaces the gas up the metering rod tube emulsion tube whatever you want to call it into the throat of the carburetor the exact same way it gets sucked up through his little straw like hole when the air is rushing over it.
There's other ways to do it but that's the two main ways.
Motorcycles have typically used a different approach and some of the small newer handheld blowers and trimmers , especially the last of the two cycle ones used a similar approach to the motorcycles as in an enriching situation which they would call a enriching or riching circuit but like I said most of us don't bother to consider that really its own separate thing.
Most the time you're dealing with a choke and it's not like it can get clogged up or anything. The flat / butterfly either closes all the way or it doesn't.
It is worth noting there has been a couple of tsbs and revisions over the years mainly on Briggs & Stratton but it's probably happened elsewhere where they made the butterfly with not complete coverage of the whole and they left a cutout in the corner. Often on equipment they would be holes drilled in the butterfly too so they could have just enough sealing without flooding it. Some of these I think it was the overhead valve vertical crankshaft ready start Briggs had an improved replacement butterfly that you would replace to make them start better.
It's been a few years now since I've had the situation and it could have actually been on a hot restart issue where it was two complete and the new one had the corner cut out because I think I remember modifying one to like the new style and cutting it out myself instead of having to buy one.
Regardless, pretty much we deal with the high-speed running circuit and the low speed idle circuit which on many things isn't even a concern today because for the vast majority of lawn mowers that are standard 21 and 22 walk behind and push mowers, there is no throttle speed so you're never really going to know if it runs well at idle or not unlike a rider or many of the snow blowers.
The trick is knowing that many of these carburetor designs will not run smoothly and even surge extremely like the Chinese Huayi carburetor.
It's common procedure for those of us who actually keep the original carburetors and clean them which is a very good day because most shops and most people throw away a perfectly good carburetor and often replace it with an inferior quality no name carburetor because the factory one is quite good. It doesn't go bad, it just gets clogged up.
The main jet for the primary high speed running circuit is the one that gets it to run but if the secondary or idle circuit is still restricted or clogged it will surge up and down and it will not run at idle..
The old tecumseh's were similar but not as extreme. When they ran but surged about 15 or 20% you could pretty much guarantee you needed to take out the secondary idol circuit jet and run a wire through it and blow it out well with car cleaner and compressed air. Then they would almost always smooth out to 100%.
These are the tricky circuits that you have to worry about and that's probably why that's almost of us are concerned with. The choke is fully mechanical and easy to inspect and observe correct operation or not.
Air is also obvious as I do get a number of mowers in every year that will not start and run or run properly and the only reason is the air filter is so clogged up they cannot breathe..