It turns out that an array of Cart Coins and Cart Releasers make a fine thickness test pattern and become useful tchotchkes when you’re done:

They’re a bit easier to see in the digital realm:

The trick is that they’re both eight layers thick at 0.20 mm/layer. With the platform aligned exactly right, all the objects should measure exactly 1.60 mm thick.
The blue numbers give the thickness measured across the stem, just above the hole, on each object:

The green numbers are the skirt thickness: 22 = 0.22 mm.
The platform has a tilt of 0.20 mm from +Y to -Y and is just about perfect from -X to +X.
The M3x0.5 adjusting screws under the (improved) platform, seen from the front (-Y) end of the platform:

The silicone plugs inside the springs are slightly compressed, so the springs are only decorative. The platform is rigidly mounted on the plugs, with only very slight compliance, and I haven’t leveled the platform in a few months.
Tightening the “north” adjusting screw by 1/6 turn lowered the +Y end of the plate by about 0.05 mm and tilted the +X side slightly higher:

The skirt thicknesses are now in blue, too.
Tightening the “north” screw an additional 1/6 turn and tightening the “east” screw 1/6 turn produced an almost perfect result:

The thicknesses don’t vary quite randomly, but I think further adjustments won’t make much difference: the total range is only 0.12 mm = 1.53 to 1.65 mm. That’s pretty close to the limit of my measurement ability on the plastic pieces.
Notice that the skirt thread, which should be exactly 0.2 mm thick all around, really isn’t. I’m going to see whether a two-layer-thick skirt measures a more consistent 0.40 mm.