This was cogent in 1910.
A lot more recently than that--and even today but more under the table. A lot of clubs still excluded members within the past few decades.
I'm sure there are still cohesive groupings of WASPs, if not large ones or effective at gatekeeping major institutions. --Still a meaningful trope, of course. But to bring it up to date you'd have to diversify, and include, for example, Indian social and professional-recruitment patterns.
Also, I do feel that GP's take is hyperbolic even in the twentieth century. My own background is mostly German immigrants, of various religions and non-religion, and the way I've been told the story none of them faced significant resistance as they moved upward in the various academic and corporate institutions of their choices. These included NASA executives, department heads, etc.
Note that in balancing GP's accusation against WASPs I'm not attempting to address the related, but not precisely complementary, phenomenon of perpetually marginalized groupings.
It still happens more informally today, of course, but it used to be a pretty (if un-spoken) part of how a lot of WASPy organizations operated to a greater or lesser degree.