Why “buy things secondhand,“ not “buy secondhand things”?

Buying things secondhand refers to the habit of searching for and ultimately purchasing used items. As you said, secondhand modifies buying, so this construct emphasizes the deliberate buying habits of the individual.

With buying secondhand things, however, secondhand becomes an adjective modifying things, so the word is describing to the items being bought, rather than the kind of shopping being done.

The difference is very subtle. Either would be acceptable English, neither would be deemed awkward or incorrect, and the meaning of both essentially boil down to the same thing.

As for everyday usage, and what sounds more "straightforward and plain," I think that secondhand is one of those words that, when used in its adverbial form, often slides in nicely at the end of a sentence or phrase – perhaps because moving it before the object would morph it into an adjective. In fact, Merriam-Webster lists this example usage for the adverb: bought the couch secondhand; NOAD mentions: tips to avoid when buying secondhand.

You've raised an interesting question, but my native ears don't find anything awkward or complicated about how that sentence was written.