I've always felt that I never steered using the handlebars at all, always just leaned one way or the other, but now I will watch out for this counter-steering malarkey! I'm sure the esteemed Sheldon Brown is right that you should always keep yourself centred over the frame.
By the way, you don't need tread on road tyres -- tread is there to prevent aquaplaning, which happens to wheels travelling in the wet at around 60 mph. If you aren't going to reach that sort of speed (you aren't, are you?), you just want as much rubber in contact with the road as possible, so any tread at all will actually lessen the amount of grip you get from your tyres. Tyres with shaped shoulders are the worst.
If you are riding on any surface which is softer than rubber, though, then you do need tread.
Cheers,
Tom