Andrew, perhaps I should have been more clear that I was concerned that you would use 1x lumber for the top. I do think that with minimal support on the outside, you would have problems with that warping or splitting. If you used butcherblock or 8/4, you should be fine and have enough thickness to put a screw in. Since you will have to do a glue-up if you use 8/4, you will want to be sure that the joints are jointed nicely and probably double biscuited.
Don't even think of using drywall screws to hold any weight. Size 10 screws would be the absolute minimum and I would just run lag screws up from the footer into drilled holes. For the support braces, Number 10 screws through drilled and countersunk hole into the bottom of the counter and into the studs should work fine. Thinking more about it, since you have the wall open anyway you could even put blocking between the studs and mount the braces to that. Then you could put the braces anywhere. You do want to have the brace rest on wood and not on drywall and then wood. You can cut holes in the drywall in the approximate location for the braces and install the braces after you hang the drywall. That might be more complicated but does give some design options.