The part that a deep tread is more necessary to drive the water out from under a car tire than a bike tyre is probably true. However, bike can also aquaplane to an extent, so treads do come in handy in seriously wet conditions. Probably a softer compound helps more than tread...
One thing is certain: skinny slicks perform really badly on loose surfaces like sand or even a dirt road.

The tread does not have much to do with acceleration, but weight does. The thinner tyres that have no heavy knobs are up to 400g lighter, and that's a lot.
Knobs also increase rolling resistance, and aerodinamic resistance, too.

The widespread notion that a bigger surface grips better is entirely false. What matters is the force that pushes the two surfaces together, and their qualities. A wider tyre comes in handy only because little stones and road imperfections mess with it less. On good dry asphalt, only the rubber compound affects grip.