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They never said he was the ONLY bar, they just never mentioned another one. Aside from when Martis moved in, but he didn’t last long. He did have the only holosuites on the station.
They may not be canon, but reconstructions of the Promenade e.g. in Star Trek Online have only Quark’s.
Also, I found this picture of a Promenade directory on Memory Alpha:
It lists “Spacely Sprockets,” among other things, but no other bars.
Also, now I’ve got to go re-watch DS9:“Rivals” because I didn’t remember it at all, so thanks for that!
And Tom Servo’s Used Robots.
Noice!