St Andrew's in West Chicago was close, but no cigar. However, if memory serves, the Scottish architect, David McIntosh, hailed from the real St. Andrews.
Here is a shot of David McIntosh in 1909. He is one of the unsung heros of early Chicago golf. Cornish and Whitten did not include him the The Golf Course, yet he designed both of these courses, had a hand in renovating Bendelow's original nine and possibly designing the 2nd nine at Glen Oak as it made the transition from Pickwick CC in the early teens before Langford's major redo in the 20's, designed Fort Wayne CC (with Langford??), and most important in my book was Langford's mentor and golf instructor while growing up at Westward Ho where McIntosh was the pro before moving to Glen Oak. The Cog Hill website says McIntosh owned Oak Hills a course in nearby Palos Heights where the Coghill brothers first played golf in the Electrical Workers union golf outing.