From a manager's perspective, the root of his frustration wasn't that easy to solve. To "scale" the team, it was necessary for the 3 guys to take lesser work and train few/more of other sysadms so that they can delegate their work. However, their work was so pressing, they hardly had time/chance to teach others. (Also individual personalities may be different...some people are great at their work, but not so great at teaching others..some don't want to give up some power..some potential successors may not be willing to take up their responsibility etc..all of this increases the problem).
My guess is that your IT dept is in a similar situation. Explain to your manager(s) that they need to clear the traffic jam/bottleneck. Tell them to find and check the IT dept's backlog so that they are convinced. Tell that this inefficiency is potentially losing the company money. Also if your views are backed by some of your colleagues it will add weight to your words. And be patient (complaining too much back-fires, so save it). If nothing works out and the situation is unbearable..then..we'll try to find a different job?