Their English equivalent are the Life Guards, whose colonel holds the office of Gold Stick for England. On, say, the bridge at Coldstream, he could formally surrender the Stone to the Captain General of the Archers, who is Gold Stick for Scotland.
Your assertion that "some 30 British monarchs" have been crowned over the Stone is inaccurate. To date there have been 18 "Britannic Majesties" of whom just six used this coronation chair. Before 1830, a baroque throne was in use.
The gilded wooden chair containing the Stone may itself have a 12th or 13th-century Scottish provenance. The lions at the base are striking. Fitting guardians of the throne of King William the Lyon? Had Edward I commissioned it, he would surely have incorporated his own heraldic motifs, leopards, etc. Modern dating techniques might prove conclusively.