Can you grind Cheerios and use it as flour?
This would behave like a literal gluten-free flour (as in, a flour that offers no glue proteins or other binders of any kind), NOT like a complete gluten-free flour substitute. 

Also, unlike some raw starchy flours that have no gluten (eg rice flour), there is not much binding action from sticky starches that can still gelatinize, since whatever starches are in that kind of processed cereal are likely completely cooked - otherwise, the cereal would become thoroughly sticky when mixed with a liquid like milk. The only binding properties you could maybe expect would be from sugar getting dissolved and recrystallized or caramellized into a firm matrix.

So, this could act as a filler or flavour/texture addition, but not as the only constituent of a dough or batter.

BTW, grinding up broken product (when it is still hygienically acceptable, obviously!) and adding it to the next batch is a practice sometimes used in commercial (factory level) cookie making.