That is the gist of it. The rivalry. Against Pahang, Johore, Penang, and mostly, Selangor. The sentiment started with the British army bases stationed across the peninsula playing one another, and soon rubbed onto the locals who eventually took over. This led to the Kallang Roar. Singapore was part of Malaya when the thing started. This feeling rose and peaked during the late seventies/early eighties until Peesai felt itself much "higher-placed" than the Malayan states and preferred to be associated with "first-world nations." The reality is that the skill level of the national squad remained the same. Singapore will never play competitively against world, or even other Asian nations.
There is no rivalry between Toa Payoh and Geylang, or Queenstown, Woodlands, Changi. Yes, better to scrap the S-League. Better for the youth to study, do homework. Once done with school, learn to char kwey teow.
Cheers!