I would say it's far from "obvious" the part was written for Dax.
Or at least they re-wrote it to perfectly tailor to Bashir. I loved his participation in "Birthright", he was his usual charming, occasionally a bit awkward self (as he was in the early seasons.) Not essential to the story, really, but great to see the crossover.
Hell, I love most cross-overs, anyway, and this one was well-done.
It did feel a bit odd that Riker would contact DS9 to talk to Quark, of all people. But, they wrote that to the character's strength. It makes sense that Quark would know the rumors and gossip more than Sisko.
A cute scene. Non-essential, obviously, but fun.
Hell, I wish TNG had more of 'em (and vice versa).
We should have had Geordi in "You Are Cordially Invited." Or Riker.
Alas....