I've a 7 1/2 horse Subaru/robin engine on a generator that I bought at auction.
It wouldn't start and we decided it was the float valve. I ordered a new carb because the replacement valve was $42.
Anyway it starts now but dies after running 5 or 10 seconds. When I first started it, after the carb replacement, the exhaust was black. Now it isn't as black and maybe runs a little longer and I don't have to choke it as much as before. I've played with the air/fuel adjustment and maybe that's what made it run a little better.
But it still dies after running a short while.
Any ideas as to whats wrong?
just

Fuel filter plugged? If I has one, is the Pickup screen plugged?

dirty tank. Flooding out, fuel line / air leak bad gas tank lid. list goes on. Some small engines won't run if the gas tank lid isn't air tight. some need air

I replaced the filter . I also blew air back thru the fuel line. Where is the pickup screen?
just

Pickup screen would be where the gas gets into the fuel line. If you already blew air through, it’s clear.

Bad gas tank lid. Maybe. there was a gray disc that turned inside the cap. I've been trying to get a replacement but can't find one.
I took the cap to the local John Deere dealer because the cap looks like one of theirs bad was told that the dicc which was in pieces was just for ventilation and he thru it away and he gave me a O ring.
Anyone know where I can get a replacement gas tank lid for a Homelite LR4300 generator with a Subaru Robin EY28 engine?
just

Collapsing fuel line due to either a partially clogged filter or a decayed/soft fuel line. Sometimes a collapsing vacuum line ffom the crankcase to the fuel pump can cause the same symptoms.
You won't notice the collapsed line when the egine is not running, but when running at full load it will then collapse cutting off fuel supply.

Also, a kink in the fuel line, especially when hidden inside the fuel tank will cause the same symptoms.

Gravity fed. I'll check for a collapseing fuel line but I don't think so.

Black smoke is a classic sign of running too rich. Too much fuel.

Check your air filter , even maybe run it for a bit with the air filter removed.
Check proper choke function , fully open. Check carburetor fuel to air at main jet. Possible wrong carburetor for application.

Google 'black smoke- small engine' for more info/tips.

I know excess gas makes black smoke, oil blue. Check muffler, sometimes engines get laid on their side and fill the muffler up with oil. Is the air filter clear? I have had engines brought in that the air filter was so plugged it would pull air thru crankcase vent. Ran terribly for a few seconds then quit. Thing about single cylinder engines is, every little thing will stop em from running. Goodluck, Godspeed.

Might pull spark plug and spin engine to make sure it turns super free. To rule out generator isn't binding and loading it down. Or something iffy in the engine.

First things first....give it a basic tune up,to .ake sure it's all in proper working order. You replaced the carb, so now replace the fuel filter if there is one, the spark plug, and the air filter. Don't assume any of them are okay. Now, if it still runs like garbage, find out if it's running rich still, or if it's running lean. Get some brake clean or carb cleaner,and while it's running, shoot some short blasts down the carb intake. If it runs worse,it's rich. If it runs better, it's lean. That'll give you a good place to start, and a give you an idea of a diag path to follow.

maybe fuel sediment bowl on bottom of gas on /off valve has crud in it or pluged screen in gas valve.

Air cleaner is clean but its not on the engine. I did not replace the spark plug. just cleaned it. It was black with carbon. It showed no burnt electrode.
As far as the gas cap goes it dies way before the gas tank could run out of air , which would happen every time it takes fuel, so I don't think that's it.

I had an old Snapper leaf blower I was about ready to junk but I tried one more place and they put in a new fuel line pick up tube and fixed it right up.

Have you asked Homelite? Done a Search on Amazon?
Don’t forget jackssmallengines either.

Does it run a butterfly governor ? If so make sure the linkage is hooked up right.

Mud wasp nest in the muffler???......jk

That is a 4-cycle engine isn't it?