Don't have any references about elk, but I have little doubt they did do elk calls, but probably with the throat, not a mechanism.
The Yahi Indian Ishi, found in 1911, was described as making the sounds of deer with his throat to draw them close to his bow.
I can't find the reference right now, but I have read of Indians making the sounds of turkeys to lure settlers to them for an attack, and of one Indian fighter who used the sounds to locate and kill the Indians.
In the battle of Saratoga, 1777, Col. James Wilkinson: “I then turned around to regain the camp and report”¦ when my ears were saluted by an uncommon noise which I approached and perceived Colonel [Daniel] Morgan, attend by two men only, who with a *turkey call* was collecting his dispersed troops.” Throat or machine?
And turkey hunting, 1760: ".... one Josiah Basset --- was creeping in the Bushes on his Hands and Knees, in order to get up to them; --- Jesse Baker having started one of the Turkey it flew and alighted near the Place where Josiah Basset was, who had on a dark grey Bearskin Coat, and had taken off his Hat and Cap, and his Hair being short and black, Baker saw the Glympse of him through the bushes, and his Hair and Parts of his Coat resembling the Colour and having as then seen the Appearance of a Turkey, Baker hesitated whether he should shoot or not, when Basset made a Noise, imitating the Cry of a Turkey, which he did very exactly, when Baker fired, at the Distance of 6 Rods, and shot a Ball through his Vitals, and sundry other shot into his Body, he gave one Scream, and the Company run up, but found him dead, one Hand only having a little Motion."
I've hunted geese with a young man who could make goose calls with his throat better than geese can.
Spence