The body does indeed produce glucose (from amino acids) all by itself, as a fuel for a variety of metabolic processes.
There's another way to look at the dawn phenomenon too though. Yes, the body will manufacture glucose, but it will also de-activate insulin in the early morning which will also cause the blood sugar to rise. This, in my view, is a better way to think about what the body is doing to ensure a supply of calories for metabolic fuel first thing in the morning. With low levels of circulating insulin, the body is able to access fatty acids for fuel, which it can't do when insulin levels are raised. Perhaps this makes more sense from an evolutionary perspective. Raising blood sugar may have provided a little fuel for the body before food could be found, but not enough to support strenuous activity for any length of time. Accessible fatty acids however can provide sufficient calories for days or weeks without dietary calories. They are a much more abundant reserve of cellular fuel than glucose in fact.
All the best,
fergus