Lindon

[The Might]

You seem to have forgotteon one key aspect while making your first assumption - the Elendil-stone was never kept at Mithlond, but in the Tower of Elostirion, tallest of the 3 White Towers that were build by Gil-Galad for Elendil. These towers were only guarded by the Elves.

As far as Gondor and the Hobbits are concerned I still must strongly disagree.

Firstly about the Hobbits, I believe that this quote from "Concerning Hobbits" should prove most informative:

Three Elf-towers of immemorial age were still to be seen on the Tower Hills beyond the western marches. They shone far off in the moonlight. The tallest was furthest away, standing alone upon a green mound. The Hobbits of the Westfarthing said that one could see the Sea from the lop of that tower; but no Hobbit had ever been known to climb it. Indeed, few Hobbits had ever seen or sailed upon the Sea, and fewer still had ever returned to report it. Most Hobbits regarded even rivers and small boats with deep misgivings, and not many of them could swim. And as the days of the Shire lengthened they spoke less and less with the Elves, and grew afraid of them, and distrustful of those that had dealings with them; and the Sea became a word of fear among them, and a token of death, and they turned their faces away from the hills in the west.

Indeed except Sam, Frodo and Bilbo, who passed over the Sea, and Merry and Pippin, who at least went to Havens, there are only two other known examples of Hobbits that went to the Sea.
Firstly, Basso Boffin, who went to the Sea in 2795 and never returned to tell the tale.
Secondly, Isengar Took, who was said to have gone to the Sea in his youth.
As such, there is little doubt in my mind that Hobbits had almost nothing to do with the Elves of Lindon.

And the Gondorians are only said to have traveled so far in times of need, such as the time an army was sent to help fight off the forces of Angmar.