First of all, these engines are about as simple as you're liable to run into today.
It appears that the plugs are either new and/or have been cleaned and gapped properly, and that the fuel filter and air filter and prefilter (if it has one) have been checked and replaced if necessary.
Issues with engine timing, and "emission controls" are so rare as to be not of consideration.
If the engine struggles to start but then runs ok once started there are only a couple things that it CAN be.
If it appears to run properly once started, the valves-(even if sightly misadjusted), are set closely enough.
Again, if it appears to run properly once started and has reasonable power, the compression-(even if one cylinder is lower than the other, or both-are a little low) is close enough.
If it starts AT ALL, the ignition is working ok-AND if it runs ok after starting, the odds of it being an ACTUAL ignition issue are effectively zero. (If its cranking too slowly to generate a consistent spark while attempting to start that is something else entirely.).
So that leaves the fuel system.
Since it DOES start, and it then apparently DOES runs ok-with reasonable power once started, it is getting AT LEAST enough fuel.
In fact, it is likely to be getting TOO MUCH fuel while attempting to start-as evidenced by the "popping".
If someone doesn't have the skills, tools, time, or inclination to clean and set a carburetor up properly, it makes sense to either take it to someone who does-OR (if determined to "solve" the problem themselves), to start with a clean slate.
In this case, since other reasonably likely possibilities have effectively been eliminated from the equation, that means trying a DIFFERENT KNOWN TO BE GOOD OR NEW carburetor-one that HAS NOT been tinkered-with.
The old one is (by now) likely to either be in need of a proper cleaning, is misadjusted, or has worn parts.
The latter includes, (but is not limited to) a float level that is too high from a float bouncing so many times it is no longer in the proper position, OR a worn needle/seat combination that allows too much gas to be sneaking into the carburetor (in SPITE of a float that may still be set properly), OR a worn throttle shaft and/or a worn throttle shaft BORE in the carburetor's casting (either of which can allow enough UNMETERED AND UNMETERABLE air OR fuel OR BOTH to sneak past the throttle plate WHEN ATTEMPTING TO START, (by distorting the fuel to air mixture from what it would be if such excess clearance was not present).
Any single one of these conditions can easily make an engine difficult to start-and in combination even more difficult.
Very true but it seems in this case he just isn't getting enough fuel because it was stated before that he sprays some gas or starting fluid etc into the carb it fires right up, if I am remembering the earlier post correctly.
It is, however, possible that there is another minor air leak somewhere between carb and intake or intake and engine or even the throttle shaft that could be letting air in thus reducing the effectiveness of the choke but it's fairly unlikely.
We often get into the mentality of saying this or that can't happen when in reality if you do this long enough you'll see a decent number of things happen right in front of you that aren't supposed to be able to happen or "never" happen.
You're right about the troubleshooting though as far too often people don't largely think these things out or just don't understand how the systems work and they replace parts hoping and grasping but in the end they're just wasting time and money.
I'm not in real agreement about the carburetor though because half the time when people buy these eBay and Amazon carburetors it's not really a fact that it's a known good carburetor. New doesn't always mean good.
Also, pretty much 99% of these if not 100% of these engines unless you happen to have something old enough to have a Tecumseh on it and those weren't common on these machines....does not have any float adjustment whatsoever because there is no bendable tang like the Tecumseh had on both the brass and the plastic version of the float.
Everything breaks from the overhead valve engines and even the flat l heads on the riders for a number of years before that and all of the new Chinese built engines etc have a plastic float but no adjustment of the float height and none of that ever gets out of spec.
The needle and seat leakage that you speak of can occur and often does on older machines and can occur as early as 8 years or anytime if a speck of dirt or hair gets in there but that almost never results in an engine that won't start or run properly..
At worst it could flood out or get a plug wet or even hydralocky cylinder but after removing the plug and cranking it over a few times, it will start right back up and run fine. The problem with the leaky needle and seat almost always occurs and presents itself as flooding after it sits and not needing any choke to start and fuel migrating over into the crankcase and diluting the oil and the gas tank being much lower than it was when you parked it or even empty.
Once one is found to be doing this then it certainly should be fixed because the workaround of the fuel shut off never works long-term because you eventually forget to turn it off and you flood things out again.
I just don't believe in replacing parts. I only do so when I absolutely have to.
I would rather keep an old original carburetor on a Briggs single OHV and put a new needle and seat into it than to put a brand new carburetor on it whether it be a Briggs & Stratton way over price carburetor or a generic cheapy readily available online.
They sing so rarely give any odd running conditions as long as you clean them out well so the only time I would ever even consider another one would be if I had some very weird running issue I couldn't put a finger on and then I would swap one from another perfectly good running engine so I did know absolutely for sure that that carburetor was perfect.
I've done this twice in the past 15 years.
Once was on a Briggs overhead valve single like we're talking about on a used mower I purchased that would start and run perfectly at idle but would not rev up.
Now it's been over a decade so I don't remember exactly how it acted I don't think it popped or spit or anything like that but it seemed more to lean out of fuel but again I don't remember exactly.
It must have acted like a lean condition because I would have probably sprayed some gas into the throat while revving it up to see if I could improve the situation but I did swap a carburetor onto it just to make sure and it ran exactly the same so that ruled out carb and fuel.
I also remember checking the valve adjustment and such but I didn't bother to do any more digging. It's possible the engine had been apart and the camshaft was installed a notch or two off or something like that.
Like with my other post, it was far quicker to just throw one of my perfectly good running engines on that machine which is what I did and it was ready for sale that day and sold it in the next 3 days. It wasn't worth taking the time to disassemble that engine and put new parts into it when I had funny of good used donor engines.
I was so busy after that I don't think I ever even tore that engine apart to look inside of it. Now, I don't toss anything until I scavenge all the parts I can off of and out of them..
I did this once again on one of the newer Briggs small push mower overhead valve engines the first year those were out.
Had that one that was surging badly. A number of them do that from time to time but this was quite extreme and I had already cleaned out the carburetor. So I decided to swap another carb that ran perfectly onto it and did the exact same thing so obviously it wasn't the carb.
I was able to increase the speed on that one which is a little low and it smooth out to acceptable levels but still not as good as I would have liked. Really never did figure out exactly what was wrong with that one but that was in the first round of them I ever saw. Haven't ran into the situation again.
I do like to understand how these things work and I love to troubleshoot more than I actually like to fix.
