Could be something along the lines of a blood sugar thing too.

Your average breakfast options tend to be pretty carb heavy. If you're loading up on protein in the morning you're going to have more sustained energy, you're going to be full for longer, and you're less likely to be on a blood glucose rollercoaster all day long. If we presume that you're battling under the conditions of limited executive function, not having to push through the sugar crash or strategically deploying willpower to avoid snacking on sugary food when your blood sugar dips would have to make your day less mentally taxing.

Maintaining a decent protein intake daily is a good idea for the amino acids as precursors, as you have mentioned, and if anyone is dealing with debilitating executive dysfunction or really impaired appetite due to meds then it's extra important that you focus on maintaining your protein intake for that reason, so a protein shake daily is a really good idea imo.

Edit: just wrote out a comment on my take on supplements and realised that I overlooked creatine in this reply. Does your shake contain creatine? If so then that would also be something to consider as the reason for why you find it beneficial, especially if you're veg or you experience depression.