"Given a choice of options, I can always try all of them and see which one gives the same result as Automatic. It wouldn't be giving anything away to reveal which option choice was used.":
But the choice will depend upon the particular differential equation, so still the documentation could not tell you in any general way which of the non-stiff Adams method or the stiff Gear backward differentiation formula is used for the explicit, or default, option Method -> Automatic.
Moreover, even if you find through experimentation that an Adams method was used, it won't tell you which order Adams method; for that you'd have to do additional experimentation (including, quite likely, implementing the particular order method).
In principle, Mathematica could be enhanced with an option that, at evaluation time, reports the choice. But that's a different level of functionality from what Mathematica currently has, and it could unduly complicate the underlying code, with unwanted effects on efficiency.