I recall seeing a video of a (Russian?) guy who had a car with a clear valve cover on it...and he started it in below zero conditions. It was surprising how quickly oil got to the top of the engine, even in those conditions. Something like a few seconds.
I would think that a bike or scooter would be fully oiled in a matter of 5-10 seconds, if not sooner? But, yes, I always idle my bikes a few minutes before riding as I gear up, first, then do a quick pre ride check of tail light, brake light, headlight and all turn signals. AFter an oil change I'll idle it for a minute or two before re checking oil level.
When you say you haven't been idling yours after an oil change, how quickly do you ride away after starting?