Does it stay running with the throttle partly open? When you pulled the carb and caught the fuel, had it just quit running and was the bowl full of fuel?
I believe that sled had a Mikuni VM30 on it. It should have a starter plunger instead of a primer, but someone may have converted it. If it still has a starter plunger, then leave it on until it runs rich and the drop it to the middle setting, and repeat to off when it runs rich again.
Can you post some pics?
IIRC the pilot circuit consisted of the pilot jet line from the bowl which connected to three other passages . . . 1 - the pilot air jet from the air horn, 2 - an orifice under the slide (bypass), and 3 - an orifice right after the slide (on the engine side). Looking closely in the venturi you should see 2 little openings in the bottom (must raise the slide to see the 1st one). All three of these need to be clear for proper operation. The flow of air and fuel actually changes direction in #2 (bypass) when the slide is opened (instead of air coming in #2 (as well as the air jet) and fuel going out #3 - fuel comes out both #2 and #3 when the slide is open)
To check your floats just fill the removed bowl with gas and set it on the bench to see if the floats drop over time.
Do check your pipe. I filled one with enough fuel to blow it completely off the engine once when I was being careless.
I did have a weak pulse cause one to quit pumping fuel when the engine warmed up. But it sounds like your issue is right after starting.