Apparently there are at least two hellmouths in the world, and almost certainly more. Greek mythology has several stories about people, the most famous being their champion Heracles, or Hercules, for one reason or other descending into Hades (and hopefully returning, but not always). Norse mythology has stories about Loki, the god of mischief and evil, being confined in hell, guarded by an army of trolls. When he finally escapes, it will herald the Ragnarok, or apocalypse. (Sound familiar?. I've always wondered if the First Evil was patterned at least slightly, on Loki) There's the Christian story in Revelations about the minions of hell being released, bringing about another version of apocalypse.
We could tie Whedon's in with those and other similar existing stories to form a system whereby the hellmouth, by whatever name, is an actual existing place with several possible points of access scattered throughout the world. If the forces of evil manage to break through at any of those spots, a champion, be it Slayer, Hunter, Guardian, Protector, or my personal favorite, Avenger, is dispatched to either reclose the opening, thus keeping evil at bay and mankind safe.
That might also explain which Potential is Called. Whoever happens to be the one closest to the existing, active, hellmouth is the one Called. How does that explain Kendra and Faith being so far away from the Sunnydale hellmouth when they were Called, you might ask? Maybe there is a hellmouth in or near Jamaica, and Faith being born in Boston is no assurance that she wasn't stationed in Cleveland.. The Powers, realizing that Buffy was still alive and well at the (we assume) most active hellmouth, simply saw to it that an alternate location was also protected. It might have even been the Powers who actually manipulated the circumstances surrounding Buffy's Slayer career with the ultimate intent being the formation of the Slayer army, the final great battle, and the at least temporary expulsion of the supernatural element from our universe.
All of which seems to have rambled away from the original question. Who fights, or more accurately, fought, the demons in and around Cleveland before the Slayer team under Wood's command came there? It's doubtful that evil had simply been given free hand to run rampant throughout the city. Probably groups of demon fighters like Gunn's crew in Los Angeles, and I wouldn't be surprised if the local police had some sort of demon SWAT team, as well as even another group like the Initiative. If we ever see more detail about the operation at the Cleveland hellmouth, we may find out for sure.