Bill,
I agree that participation was probably less than the NE, maybe just because of the damn heat, more concentrated wealth vs. NYC etc. (although there are some early public courses in TX, too)
Hard to know just what effect different wealth levels (as wealthy as Texas Oil was, it still never matched the NYC area) too much heat, Cowboy/Wild West culture, including self reliance mentality, anti-Yankee mentality, poor turf, etc. but I figure they all came into play somehow.
In a lot of ways, Texas was behind the NE, so when the Golden Age came to the NE, it was likely to come to TX later, which of course put it in the depression, WWII, etc. The general expansion of culture in TX probably came after WWII, and we seem to have more RTJ courses, etc.
Again, just a loose theory, more fun to muse about, but someone will probably put some real research into it.