[capnward]

A full rebuild of the carb, which is a replacement of the jets and other internal parts, is unnecessary as long as your existing parts are clean and installed correctly. But it won't hurt. With ethanol fuel the rubber tip of the float valve needle may degrade over time, in my opinion. The rebuild kit you get from Moyer has a small tapered wooden dowel which is used to dress the float valve seat. Should be no need to remove the seat.
Since you are taking the carb apart, I suggest you remove and inspect the tiny idle jet, although that is not causing your non-starting problem. File down a screwdriver so that it fits the slot in the jet snugly, and doesn't damage the threads in the carb. Once you have removed the jet, you can hold it up to the light and see any crud blocking it. It doesn't take much to mess up your idle. Then run a thin wire around in it to clear the crud. Spraying cleaner through it isn't enough. You wont see any blockage unless you remove it.
But I suspect your problem may be idle mixture adjustment. What you called the idle jet is actually the idle mixture adjustment needle valve. Page 4-3 of the Moyer manual shows this difference clearly. If you had it turned further out than 1 1/2 turns, it was adding too much air to the mixture, so it wouldn't start. Same effect as the choke not closing completely. Make sure you have a tight fit at the gasket where the carb joins the manifold. A new gasket helps with this.
Turn your adjustable main jet in all the way, then back it out 1 1/2 turns. This may clear any crud in it.
If you are one of us primitives with a mechanical fuel pump, you can use the hand bail to pressurize the fuel line all the way to the float valve before starting, and it will start sooner. If you have a fuel pressure gauge, you will see when the line has pressure. But you can feel it pressurize when the bail goes completely slack. You can hear the fuel filling the system as you pump it by hand. Then the sound goes away when it's full.
+1 on fixing the scavenge tube right away. You do not want it dripping into the bilge.
Keep up the good work. You are almost there. Let us know how it goes.