Bit of a mystery. If it was the float valve leaking, fuel would be coming out of the overflow tube.
And why just at idle?
A reversed slide will suck a LOT of fuel through the main jet at "idle" or whenever the slide is closed when running, but if it's sucking enough to dribble out the exhaust and dilute the oil... it would be unlikely to fire and your idle would be very lumpy. Plus it's easy to reverse two (or all four) slides, but just one would be something of a trick.
A fallen main jet or missing main emulsion tube could also cause a lot of fuel flow but at idle, with the slide correctly positioned, there's not much vacuum at the needle jet. It would also run really really badly under throttle.
Physical damage that's opened some carb passage to the throat? How likely is that, and which one would cause fuel flooding at idle?
Misconnection? Where? there's only one fuel line from the tank and if it was connected to a vent barb there would be major problems with fuelling.\
The situation at idle is high manifold vacuum that would suck a lot of fuel if there's a path to the bowl from the engine side of the slide. The idle circuit opens just at the engine side edge of the slide, but you'd need a huge pilot jet or some sort of damage to get that much fuel through there. The needle jet is on slide centre back into the cutaway: this leaves it at close to atmospheric pressure (low/no vacuum) when the slide is closed at idle.
So no answers but a tip: if the overflow isn't dripping the float valve is not your suspect... assuming the overflow isn't blocked of course. Check the slide - cutout towards the air cleaner, and that the jets and emulsion tube are present and OK.
