Senior Administrators do not only deal with punishment appeals. If you check the staff roster, you'd see that all of us are involved in many areas of the staff team. I for example assist Osborn with leading the administrator team. A lot of my time goes to reviewing administrators, assisting them etc. Even if you put our areas aside, we are still, as the title says, Senior Administrators. We deal with investigations into serious rule breaches, discuss day-to-day operations and assist with managing the server.
Dealing with the appeals was not an issue for example three months ago, but as you said, there has been a player growth. It is not a simple process to just promote other Administrators to Senior Administrator, as the latter is a rank that is difficult to achieve for many reasons. We do our best to respond to appeals, and whilst I agree that response times are not as fast as player reports for example, we attempt to resolve appeals as swiftly as possible.