What ways can we observationally probe what is happening inside of stars as they evolve off the main sequence and thereby directly test the predictions of stellar evolution theory?
We do not need to look at the stars to understand what is happening inside it.
Let's take an example of the sun. The sun is quite simply a giant nuclear fusion reactor. A combination of chemistry and physics gives us the answers we need for the sun and most stars. We do not yet understand pulsars, neutron stars and a few others. By knowing the mass of the sun we know both its history and its future. That's how we can observe what's happening inside the stars.