A lot of things here but I'll try to cover the basics.
Firstly, I'm never a fan of replacing Parts especially carburetors because they rarely ever wear out unless they're 20 plus years old and the throttle shaft has worn into the aluminum body and it's sucking air. That's about it.
Nine times out of 10 even if you replace it with the same brand name carburetor like Briggs & stratton, the 8 to 20 year old carburetor you're replacing is probably better than the new Briggs and Stratton carburetor you're putting on!
Carbs don't really fail, they simply get dirty or need some new rubber Parts in them or float etc.
Now having said this I have used a couple of No name eBay or Amazon carbs even on riders and had good luck with them but I prefer just to use them to scavenge parts from the inside to put into the other carburetor but unfortunately not all of them are exact clone replacements and they will change stupid little things like the float or the needle valve when they don't have to.
So, I don't think you should have gone on the road to replace the carburetor because it certainly does sound like it was a fuel delivery issue and more than likely was the fuel solenoid on the bottom of the carb.
You will hear this called abs for antibackfire solenoid and you will hear it called after run solenoid or just solenoid on the carb like we're calling it..
You can't check this with a 9-volt battery or even the machine and be 100% assured that it's not bad.
The basic test is to see if it clicks or to feel or hear if it clicks when you plug the connector into it with the key on or you can actually remove it and look at it and watch it open and close cuz sometimes they get gummed up but this still doesn't alleviate two of the problems that these will commonly do.
Some, but not all have a rubber tip on the end and this rubber tip will become loose and will vibrate up the shaft that is knurled and will clog up or block off the jet I mean and then soon as the solenoid functions it's cycle a couple of times it works back into place so it will intermittently shut off while you're mowing and then restart and mess with you.
Then, there's the fact that many of them will just get old and they will get hot and break down so they'll work for a few minutes maybe 15 maybe two then they will start becoming erratic.
The way to test for this is to mow until it messes up and then unscrew it and put your finger on the hole and start it and see if it runs. Even better if you can find a carb bowl bolt or a short bolt with the same threads or even a rubber stopper at this point just to block off the fuel flow to test it.
I'm glad we're at the current day we are now because you probably got that part for under $12. If you would have bought that part over 5 years ago it was one of the most overpriced parts of lawn mowers for the past 30 years and they were over $60 to buy anywhere because it had to come in the brand name box because no aftermarket companies were copying it..
Then finally they showed up for $7 on ebay. This is a case where I'm glad to have the cheap aftermarket parts because they were hosing us on the brand name ones that should have never been over 10 or 15 to start with.
Kind of like golf carts and jet skis and boats. Total rip-off for what it is.
So, you're more than likely found your problem with the carb solenoid because these carbs pretty much either run or they don't due to being gummed up inside and there's just one main jet and if you take it out and or run wires through it and blow everything out clean and there's not stuff floating around in it the next time you're taking it apart you're usually good to go.
I think I read this was a Briggs engine but I can't remember. Those use either a Nikki or a walbro and I actually prefer the wall bro even though it's harder to pull the seat out and replace the brass seat I just find the repairs to be more durable than that crazy convoluted plastic body Nikki but they both will get the job done as will be aftermarkets.
The key to lawn mower repairs is diagnosis, diagnosis, diagnosis. Too many people throw parts at them hoping to find the problem and it's usually their third plus attempt before they actually stumble onto the fix and they're usually almost $200 into it at this point which really sucks.
There's other things that could be your problem like a blockage or restriction of fuel coming from the gas tank or if it has a fuel pump it could be sucking air on some of the lines too. I have more than once found those little tiny Japanese fake ladybugs inside the gas tank and stuck down into the first inch of the fuel line blocking it and I've also seen long strands of grass four and five inches long work its way down into the fuel line like a bundle of three or four of them preventing fuel flow.
Then, fuel line can crumble inside and deposit rubber specs inside your carb which is a self-clocking situation or you can get an internal prolapse inside the fuel line where the inner lining gets a hole in it and ends up ballooning up inside and blocks itself off inside.
I had one just last season on a push mower I think it was a Honda that had almost no fuel coming out of the fuel line with the valve turned on and when I took the fuel line off I couldn't even blow through it with my mouth so I started squeezing it and found the area that felt funny and cut it right there with a razor blade and you could see how the inner wall had pulled away and sealed itself off.
95% or somewhere thereabouts of lawn mower no starting or running issues is not enough fuel so the first thing to check anytime one diesel won't start is to get you an external fuel source and take the air filter off and spray that fuel source into there to see if it will then start and stay running as long as you spray it.
DON'T you starting fluid because it's just a waste and it's too hard on the engine.
Either put some gas in an old spray bottle or get you a can of spray carb cleaner because you can stand there and run one all day on the carb cleaner as long as you keep spraying and stopping and spraying and stopping.
