So, I pulled the carb apart. When I got to it, the "throttle cap" at the top was actually loose. Too, the floats were indeed a little off. The inside wasn't really dirty, but I went ahead and cleaned it anyway, inspecting the o-rings and gasket to make sure all was well. I put it back together after adjusting the floats (took a minute to line back up the needle& throttle valve, but I took my time and got it back together and the correct hoses on their stems.
I had to wait for the battery to finish charging, so just a bit ago I installed the battery and tried it out. The starter was trying, but no ignition. Kickstarting doesn't do it, either.
I am, though, incredibly grateful for your help and advice in this, as I now have another notch in my belt of scooter ownership.
I now need to figure out what the heck is going on. The insulator gasket is aligned correctly up/down. It's o-ring is against the manifold and the carburetor o-ring against the insulator.
Harumph:)