Brings back memories
My first sled was a 1978 Enticer 250, and I know the frustration of them not starting. Whether or not I had the same problems that you have, I don't know, but I'll share my experiences.
Mine was always a bit of a hard starter when I first got it, so I decided to rebuild the carb/fuel pump (Mine was an all in one unit, I'm assuming yours is the same). After I got it back together, it wouldn't start. After many hours of pulling on that stupid rope, one of my friends thought we should pressurize the fuel tank with an air hose to get the system primed. I wouldn't say it was the safest procedure, but it did the trick, she fired up.
It was still a bear to start. Sometimes it wouldn't even start at all, no matter what I tried. I then discovered 2 tricks that got it started every morning.
Trick #1: The gas tank had to be full, or nearly full. I have no idea what it's deal was, but there was nothing you could do to start mine if it had a 1/2 tank or less. As soon as it had a full tank, it would fire right up.
Trick #2: Adjust the idle. I actually turned my idle down to get it to start. Unfortunately, when the engine was warm, the idle was so low I had to constantly blip the throttle to keep it alive, but if I turned the idle up, it wouldn't start.
It seemed to restart fine when it was warmed up, but getting it started in the morning was the trick. Make sure the spark plug is in good shape, and you might want to try a compression check to make sure it still has enough compression. My guess is there is something up with carb/fuel pump. Be careful if you decide to tear into that carb though. The gaskets and diaphragms are likely to be very hard and brittle, and rebuild kits are hard to find.