Ferrite toroid vs E-core for high frequency transformers
Why couldn't I employ these toroids in a half bridge or push-pull
topology as opposed to the E cores in the photo?

Three reasons that spring to mind: -

It's a lot easier to manually wind turns onto a bobbin then add that bobbin to the cores compared to manually winding a transformer on a toroid.
With core-sets, you can apply gapping to give you superior performance in many cases. Of course that's virtually impossible for a toroid. (It can be done but with difficulty).
An E-core set will have more volume of ferrite and be able to handle higher saturation currents for a given floor-area of a toroid.


I've read that for a number of reasons toroidal cores make superior
transformers.

That's not usually the case.