Historically, most civilizations, including western thought, have found that syncretism is bad. It results in a shallow understanding of what it is dealing with and a hodgepodge of incompatible ideas.
Budhism (and lots of other things) was evolved and developed to be seen as a whole and in a few specific contexts. Not for mixing and matching random parts. You might get something out of it, but it won't be budhism anymore -- and, judged by budhist standards, it would be BS.
I guess it would be the equivalent of copy/pasting code, instead of properly thinking about your program's architecture.
Wha? Buddhism has a long history of evolving and mixing with local cultures, just like any religion. The zen stuff that these "koans" are either imitating or satirizing (depending on your view) comes from mixing Buddhism and Confucianism and Taoism and who knows what else.
It was not a case of "I saw a documentary/bought a manual on budhism, I am so zen now" while shopping stuff from Whole Foods.
Because those belong to a culture (with the sociological sense), whereas the second kind of use you describe is what's called syncretism, and is a quite vulgar disregard for subtetly, context, fitness and it's subject matter in general.