The PCBs for the micro-USB version arrived:

Some notes:

• The connector works. Yay.
• It's a little bit tight thickness-wise. More on that below.
• I made it a little too deep — I can make it about 1.5mm smaller.
• It's strong, even with the vias all lined up like that (later version has the vias more spread). It shouldn't break off.
• There is a bit of sideways wobble possible, but it doesn't seem to cause any connection problems.
• The width of the cutout is just right.
• The pad on the top of the connector, that was supposed to connect to the shield, is useless, and can cause a short if the plug is inserted wrong way around.
• I'm happy with the prototyping areas, I'm going to leave them like that.
• The connector doesn't necessarily look like an USB connector, so I'm going to add a label to it describing it as such.
• The staggering of the contacts, so that power gets connected before the signals, is superfluous, because they are already staggered in the plug. I will remove that.

Now, speaking about the thickness of the connector, it seems to be on the edge of acceptable. The first insertion requires a bit of force and leaves marks on the contacts:

This seems to be mostly to the thickness added by the HASL process of coating the contacts with solder. I hope that removing the pad from the other side will improve the situation here.

Speaking of the board thickness, I also made some 0.8mm PCBs for the USB-C version, and it does sit in the plug much better, while still not being too tight, so that is the preferred thickness for that kind of connector for me.