Tsumugu exists for a few reasons, actually, you just have to look at the whole picture. One is as Synnae says, to add a bit more meat to Nudist Beach, so that it's not just one guy + faceless extras. Another was to show that Nudist Beach was supposed to be regarded as something other than a crazy teacher rambling about silly shit in the early episodes, and to show that they can achieve shit and actually be a thorn in the side to Hounnoji and REVOCs, to some degree. A third reason was to make him a fakeout rival. He was never meant to be a game changer, but rather a circumstance of the game itself.
Also, speaking as someone who jumped in about halfway through the Naturals Election, the early episodes flow nicely enough when not viewed with a week between them. The over the topness and humor hook one very quickly and keep you wanting more. This is why the "filler" episodes work, actually, because they provide some extra humor and a little distraction from the main plotline for just long enough that they don't wear out their welcome. The fight club episode, in particular, is important because Ryuuko did give the Mankanskoku family a way to advance themselves, and it would have been a waste not to look into it. Not to mention that Fight Club Mako was the best thing and she needs to come back.