A good strong spark is required, particularly on a cold day. From what you say the plug is getting wet so the carb is working OK at least at idle settings.
Any signs of life at all?
My tuned 604 was very tricky to start when the weather was cold and I used to run a grade 'softer' plug which helped a lot.
As John said, the bigger the battery turning it over the better. Presumably it is turning over smartly and not suffering the dreaded starter sprag clutch problem
I also found that cleaning up the ignition sensor and 'pips' on the flywheel plus checking the gap between the pips and sensor is worth doing - but perhaps more of an issue on the early CDI system.
Keep at it. There must be a logical reason. No mouse nest in the air filter or silencer ?
"Do not adjust your mind, there is a fault in the reality"
1957 175 Sport