So, then this got me wondering; the Brother printer has been working ok, but it developed a really annoying habit: it's duplex (prints on both sides of the paper), but even when printing a single page, it would print, then pull back the paper, and print nothing on the other side. So every page would take twice as long.
I 'removed' the Brother printer, downloaded the driver-only file from Brother, and did another manual install. Lo and behold, it now prints one page as one page. The test print also started printing the heading and the printer logo in black, instead of a shade of gray that I think must correspond to the blue that comes out on the Phaser's test page.
So, print a test page, and if the "Printer Model" is "Microsoft IPP Class Driver", instead of something with the manufacturer's name in it, there's Microsoft fuckery at work that is probably giving you a less than satisfactory experience.