I always got the impression that the elves weren't really that used to company, especially not other races. The idea that a hobbit might feel uncomfortable or awkward in a social situation doesn't occur to them. And since elves are seldom ill or injured, that Frodo might still be feeling fragile would be a foreign concept also.

It's not that they're rude or inhospitable, the opposite in fact - but when it comes to the finer details of hobbit feelings, they just don't get it. After all, Bilbo's probably the only other hobbit many of them know. And in many ways, he's not a typical hobbit. Also other than Aragorn, who is also rather exceptional, most of the elves likely don't know that many Men either. Or dwarves!

It's likely the elves thought it would be nice for Frodo to be with other non-elves and visitors. I've been at parties like that, especially weddings, where family groups will be seated together, people the couple works with are together, etc. and the "miscellaneous" people kind of get lumped together.