They instead chose Eärnil, who though being still of the Royal House of Gondor was not in the line of direct succession; his great-grandfather Arciryas was the brother of Narmacil II, and Eärnil's great-great-grandfather was King Telumehtar Umbardacil.
This palantír was used by the Kings of Gondor, but when Minas Ithil fell to the Ringwraiths, Eärnil II stopped using it; not only did Sauron now have access to the network, but the palantír of Anárion had the strongest link of all seven to the Ithil-stone.