And I think we are meant to infer that Hobbits speak with an “accent” — and isn’t it mentioned somewhere by Tolkien that hobbits had lost the concept of the familiar/formal second person, so that when Pippin talks to and of Lord Denethor, he only uses the familiar mode, which leads to Gondorians concluding that he must be a Prince of his people.
Appendix F II On Translation
'This was one of the things referred to when people of Gondor spoke of the strangeness of Hobbit-speech. Peregrine Took, for instance, in his first few days in Minas Tirith used the familiar forms to people of all ranks, including the Lord Denethor himself. This may have amused the aged Steward, but it must have astonished his servants. No doubt this free use of the familiar forms helped to spread the popular rumour that Peregrine was a person of very high rank in his own country.' Pippin
is in a manner, but not in the way the Gondorians would expect, as Shire Hobbits are far less interested in rank; this could easily explain the loss of the deferential forms in the Shire, except in the Westfarthing where they used as endearments.
This section goes on:
'It will be noticed that Hobbits such as Frodo, and other persons such as Gandalf and Aragorn, do not always use the same style. This is intentional. The more learned and able among the Hobbits had some knowledge of 'book-language', as it was termed in the Shire; and they were quick to note and adopt the style of those whom they met. It was in any case natural for much-travelled folk to speak more or less after the manner of those among whom they found themselves, especially in the case of men who, like Aragorn, were often at pains to conceal their origin and their business. Yet in those days all the enemies of the Enemy revered what was ancient, in language no loess than in other matters, and they took pleasure in it according to their knowledge. The Eldar, being above all skilled in words, had the command of many styles, though they spoke most naturally in a manner nearest to their own speech, one even more antique than that of Gondor. The Dwarves too, spoke with skill, readily adapting themselves to their company, though their utterance seemed to some rather harsh and guttural.'
This section seems to provide great guidance for what is happening at the Council: The Elves, Dwarves, Frodo, Bilbo, Gandalf, and Aragorn are likely adapting to an average level somewhere near Gondorian speech. The narrator, however, I suspect renders the discussion in language more familiar to the ear of Shire Hobbits to allow the audience to focus on the story rather than the odd speech.
I think, without actually thinking it through too carefully, that the ability of everyone in ME to converse together in the Common Speech is more a literary conceit than a linguistically-correct depiction of language drift. Kind of like Star Trek's universal translator, it makes the story easier to tell.
In our world, 500 years separation makes a formerly-common language difficult to understand. 1000 years and they are different languages and mutually incomprehensible. In ME, it has been over 3000 years since the battle of the Last Alliance. It doesn't seem that even extensive trade could overcome the language drift over such a time span.
The only theory I can come up with to "explain" this would be the influence of the Elves (to whom 3000 years is a shortish time). But they are so insular, I don't see how they could be continuously keeping Westron "pure" over such a large region of ME.
We do see Hobbits and Rohirrim comparing words and finding parallels between their speech. It seems that the Rohirrim retain their own language and that this is related to the former language of Hobbits, though the latter retain only some archaic words and names and have otherwise adopted Westron as their own.
It is curious that Tolkien the philologist would employ such a literary conceit without comment.
I think the comment you were expecting is in Appendix F. A literary conceit perhaps, but with some in-world social behavioural support. If the enemies of the Enemy universally revered the ancient language this would go a long way to slowing the drift of languages away from each other. Language drift in our world is the only model we are familiar with, but it doesn't make it the only possible model.