Should I not sear and broil eye of round the way that I do filet mignon or chuck tenderloin?
Eye of round is very lean. Very, very lean. So as a steak, it would not stand up very well to any cooking method that gets it much beyond rare. And even then, it's going to be a tough piece of meat. My suggestion would be to prepare that cut differently (such as macerating it and making country fried steak or Swiss steak), or omitting it from your rotation. Your method is sound for preparing a steak, but I would limit it to:


Tenderloin
Strip
Ribeye
Chuck eye (not chuck steak, which is a tough piece of meat, but chuck eye, which is cut from the eye roll).


If you want cheaper beef, the last in the list is your best bet.