Gandalf's comment about Frodo

[Pervinca Took]

True, but he had already been a Ringbearer, and taken probably at least as much hurt from being one as Bilbo had, if you consider the effects of the Morgul-knife. Of course, it never needs to be considered, because he does assume the role "officially" at the Council. The Wise would have been very culpable, I think, if they'd left Middle-earth without asking for an option for him to join them given all that did happen, but I think they would still have had some level of responsibility to him even if he hadn't taken on the quest. The wound alone would not have been enough to make Frodo want to leave, I think, but still. Perhaps the Wise would have left the Last Homely House open to the Ringbearers as a place of peace and refuge, even if with Elrond and Vilya gone, its healing virtues would have been much lessened (and Bilbo was already resident there, of course). Presumably there were still healers with some of the skill that Elrond had - his sons, and some of the Dunedain, and the remaining Elves themselves.