This is just a comment but in community wiki format. Most studies of semisimple (or reductive) algebraic groups and finite groups of Lie type emphasize counting the number of classes of various elements with a view toward representation theory. So it's tmportant to consider motivation when counting elememts.
Concerning references, much of this goes back to Steinberg. A short summary of further work is given in my 1995 AMS book on conjugacy classes, e.g., section 8.9. In spite of their misleading title, classes are the subject of a paper bu Peter Fleischmann and Ingo Janisczak here. Note too Steinberg's theorem stating that the set of regular semisimple elements in any semisimple (or reductive) group is open and dense, consistent with your observation: see 2.5 in my book.