I see picture frames held together with glue, but I'm wondering if equally robust and aesthetic results could be obtained using flat L-shape angle brackets? What problems am I going to run into using this method?

I see picture frames held together with glue, but I'm wondering if equally robust and aesthetic results could be obtained using flat L-shape angle brackets? What problems am I going to run into using this method?

You can certainly do this if you wanted, and actually it wouldn't be a bad way of reinforcing some large or heavyweight frames — Strong joints to use when making custom frames :-)
But you wouldn't want to do it on every frame, as I mention in the same sentence in the previous Answer you can do the same using just a triangle or chevron of plywood (or hardboard) that is tacked or screwed to the back of the frame corners in just the same way. These would be cheaper of course (potentially free if using up scraps), and perhaps more importantly would have less play in them (or none at all) so your joints would be made more secure.
This is assuming the mitres are already glued of course, I can't think of a good reason not to glue the joints.
It's more work and may not be the solution you are looking for, but you can achieve a much stronger corner with equally appealing aesthetics by using a mitred half lap, such as: