Has this become a thing in crossword puzzles? I've never seen this done before...
For about 30 years, yes. At least they've been increasingly common since then.
Here's a list of all rebus puzzles published in the New York Times.
They almost always appear on either Thursday or Sunday. People who solve the Times puzzle regularly understand that Thursday tends to be "gimmick day" where you might see a rebus or some other unusual trick. There are, for example, "Schrodinger puzzles" (link goes to examples) in which one (or possibly more) answer exists in one of two possible states, because the crossing answers through the word are acceptable answers with 2 possible letters. The most famous of these was surely a Tuesday 1996 puzzle in which the middle spaces could plausibly read either BOBDOLE ELECTED or CLINTON ELECTED depending on your choice for crossing answers. As an example, "Black Halloween animal" could either be BAT or CAT, and your choice gave you the first letter of Bob or Clinton. Most Schrodingers appear on Thursday, but that Tuesday election day was an exception. Other gimmicks involve unchecked squares (squares for with there is no crossing word, and thus must be inferred some other way).
In general, for the NYT, the Monday puzzle is easy, the Thursday puzzle has a gimmick, the Friday and Saturday puzzles have no "theme" and a lot of open space (fewer, longer entries), and the Sunday puzzle is large with usually a light, humorous theme but occasionally a more complex one like the PEA rebus.