Doxygen can generate a list of all class members, including those from all ancestors. If you wanna do that checking header files you'll have to keep jumping between files going from parent to parent class.
I hope he meant something else since he said "nowadays". Doxygen is ok if you haven't setup your build environment. But for when you actually are in the code (and somewhat familiar with it) I very much prefer just reading the header files than switching to a browser.