There's no such prioritization on your body. Everything happens concurrently. It's constantly turning glucose into fat and fat back into glucose. It's constantly breaking down amino acids and reforming them. Different processes will just occur at different rates depending on the concentration of reactants and products. So going from amino acids to glucose happens a lot slower when you have other energy sources, but it still happens.

I also don't see how your comment supports your claim. Even if this prioritization takes place, the protein you consume still serves the purpose of being an energy source. That's very different from being unable to burn amino acids for energy at all as you said earlier.