Tom,
I think it is far more than the question of what were the best golf courses in the world at that moment. I think the better question would be which courses were hosting the more important golf tournaments.
One year after Shawnee, his first course, opened, Tillinghast and Worthington (Shawnee's owner) started the Shawnee Open in 1912. The best players, amateurs and professionals alike, descended on Shawnee-on-the-Delaware to play. from the first tournament through the mid-twenties, the Shawnee Open was considered one of the most important tournaments in the land, achieving what we might consider near-major status today.
The next year, with this one course under his belt, Tillinghast was on the road to design Brackenridge Park, Fort Sam Houston Golf Course, a redesign of the San Antonio Country Club and out to San Francisco to close the deal on the San Francisco Golf Club.
His early success and popularity as a preferred designer is owed to his consummate talent in promoting himself, and he did this through the Shawnee Open. The popularity of the course and praise for the designer was immediate and received coverage from sea-to-sea as much as anything that Tilly did, because of those who played in it and returned talking about it.
Baltusrol was also one of the world's great courses at this time having hosted several national championships already, and yet in 1918 they agreed to scrap their great course so that Tilly could design and build two new ones.
The man definitely knew how to market his services.