OP here, reporting late on my own question and confirming what the more recent posters have said. I did eventually buy a new SP and with minimal jiggling can lock it into the SP3 dock without an adapter. It also works easily with the SP4 adapter. To my surprise, if I put the new SP in the dock with the type cover attached, there is no need for an adapter at all; the Surface blade connector slides right into place. I don't see what part of the cover adjusts the tablet's position, but it seems to work more smoothly than locking down the tablet without the cover attached.
I very much prefer the older Surface dock designs in which the tablet is supported and locked in place at an angle. The newer brick dock is serviceable, but just makes for another wandering piece of clutter on a desk.