Despite the Emerald Chain's bluster, they seem more like competition than enemies. After watching The Ready Room, I got a different perspective. Janet Kidder, who plays Osyrra, said the Emerald Chain views itself as a society that's worked hard to earn its place in this new landscape and wants to be recognized. That's why Osyrra scoffed when Saru said that Ryn wasn't wanted by any recognized societies. Kidder said that the Emerald Chain is jealous of the Federation. And that's when it clicked with me.
Now I have a sense of what the Emerald Chain is. They're loosely like another version of the Dominion. An organization of worlds that's opposed to another organization of worlds (the Federation). The only difference is the Emerald Chain isn't as far along as the Dominion were. And the Federation at its peak probably could've mopped the floor with the Emerald Chain. But with the Federation decimated (literally, since it's a tenth of the size it was: 38 member worlds down from 350), the Emerald Chain sees itself as on the way up. The Federation not acknowledging them as legitimate or "recognized" fuels them even further.