I know little about timing (I'll have to learn, I know
) but this thing doesn't seem timing related to me.
I'd bet on something related to fuel flow, so yes, the carb could be involved, but try with simple things. I'd check:
1) Fuel pump
2) Fuel filter
3) Fuel lines (as a last resource).
If it takes a lot of cranking before firing up in the morning (or after it has been sitting for a while), a weak gas pump could be the culprit.
For example, I experienced similar symptoms (ages to fire up and sputtering after stops) and I found out the diaphragm in the pump was cracked. This also led to gas mixing up with engine oil (check your dipstick if it smells like gas and the oil appears thinner than it should, you're there).
A clogged fuel filter could be another problem, and also some spark plug wire not firmly seated on the dist cap or plug sometimes might have such effects.
In the end, start with the simpler things and step up in complexity. The rebuilt carb is the last thing to investigate, and if everything else turns out to be OK, then you're going to claim your warranty with confidence...