Stumbled across this comment on Matthew 12:16 from The Lutheran Study Bible:
As on previous occasions (e.g., 9:30), Jesus downplayed His ministry of physical healing so that His disciples would better understand that He came, above all, to bring spiritual healing.
I’ve mentioned elsewhere that a Brazilian colleague has written that the miracles are parables, too. So perhaps Jesus didn’t “downplay” his healings so much as attempt to interpret them for the people and, when he couldn’t do that, sought to keep them from being broadcast without proper context. With that tweak, I do appreciate and agree with the sentiment of the quote.