Hi guys, sorry to take so long in getting back to this (and for not answering the question when I first posted). It's been a long and very exciting day as I will hopefully be able to share tomorrow in a new topic with a major announcement from the Tillinghast Association!
You'll have to wait.
The answer to the above question and to all the suppositions is that, just like Agatha Christie wrote in the Murder on the Orient Express, you all did it... that is, you all are correct and incorrect.
Let me explain.
The Fenimore CC was formed in 1920. The core membership purchased the 40 acre estate of the Baron Eugene Reynal. They would add more property to this, using his elaborate mansion as their clubhouse, and by the spring of 1921 finally have enough land to build an 18-hole golf course and an additional 9-hole short course. They hired Devereaux Emmet, who was then at the height of his popularity and had built many fine courses in the Metropolitan area, to design these.
The courses opened for play in 1922, yet almost immediately the membership began to complain. They had a fine course, but 1922 saw the official opening of the 2 new masterpieces at Baltusrol, the plans announced for what everyone recognized would be 2 incredible courses literally just down the street at Winged Foot, Fresh Meadow which would soon host 2 national championships open just over the bridge on Long Island and another course of great promise opening on the other side of town, the Scarsdale CC, due to open in 1923.
The mebership was jealous and, even though the Emmet course(s) were fine design, they were limited because of the scope of the club's property.
Still, give them credit, they bit the bullet and took the one common denominator in all of the above courses, they were designed by Tilly, and hired him. They next bought more adjacent property so they handed him over 240 acres to work with.
Tilly basically gutted the existing course(s) and started from scratch. I am uncertain if there are any holes or features that are leftover from the Emmet work, but if so it amounts to very little.