so a worn out threaded tire can still have a very strong grip on dry roads similar to slick tires???

Not exactly... there are a number of other issues hear that relate to grip. Ultimately, street tires are not very similar to racing slicks, with tire construction and compound having much more to do with the added grip than the tread pattern (and tire shaving has more to do with minimizing tread squirm than increasing contact patch - which can also increase the longevity of the tire since less heat is fed into the compound as a result of the reduced tread squirm... and tires with no tread pattern are not tires that would be shaved).

A tire than has worn itself down past the tread so that it is putting all of it's contact patch to the road will grip better on a clean dry road than the exact same tire with less contact patch - but probably not better than the same tire when it was new (age and wear will have taken at least a small amount of grip from the tire, which would be more than the extra contact area will give it now... and for a performance tire, this would be an even larger gap).

On a wet surface, the tire will be very prone to aquaplane and will have little to no grip. It may also run into similar problems along a windy road near the beach where sand and dust blow onto the road way.

If you are wearing down to the chords, they tire may still offer some grip - but you are getting dangerously close to having a major failure, and one that could come about quite unexpectedly. With street tires, when you have that little tread, it is time to replace the tires. If you have chorded the tires, it is past time to replace them.

a worn treaded tire will have good dry grip if it is worn evenly and there is still rubber to grip (no threads!) it will be dangerous in the wet though. a good example of this is when racers who must use dot approved street tires shave the tread down to minimum depth before racing on them. they wont last long but offer better adhesion and less squirm, or tread wiggle. hope this helped!