That the fuel pump runs a lot is a bit puzzling. There are plenty of reasons that it would do that, but I can’t think of one that is consistent with the engine running fine once you’ve persuaded it to start. If fuel delivery is inadequate then you’d expect the engine to struggle when underway. Likewise, if the pump’s running because a carb is flooding, you’d find it running on 3 cylinders at low revs and lousy fuel economy.
As Darrsi says, carb icing isn’t likely to be the cause… it takes several minutes to build up enough ice to block the jets.
A weak spark could be the cause of the engine dying. The first thing that happens when the engine fires up is expose a cold spark plug to a lot of steam, so you get some condensation which can cause the next spark to track across the surface if it’s a bit feeble. The engine then dies and when the starter motor has turned the engine over enough to blow the dampness away, the plugs have retained a bit of warmth so the next time it fires the condensation isn’t bad enough to kill further sparks.
Trim the HT leads by 8mm and fit new NGK SD05F plug caps if that’s not already been done.