I have been looking through the cheap microcontrollers with USB support, and I have narrowed my list to two, with maybe possible third.
The first candidate is the venerable ATmega32u4. It has been around for a long time, there is a LUFA library for handling USB with some examples for the MSC device, and I have a whole bunch of cheap Pro Micro development boards I can experiment with. It also has a lot of GPIO pins, so it could also be useful for projects like #LAMEBOY for handling extra functions, like the buttons and backlight.
The second candidate is the SAMD11. Much more modern, cheap and comes in an easy to solder package. It also requires fewer passives — I think it can even work without the crystal. There is also a MSC device example for in the ASF3 library. It is however more of an unknown for me, as I have never before programmed any. I got a #DiXi board to experiment with it.
Finally, a distant third is the good old STM32F103 chip. It comes on the "blue pill" dev boards, and it also has some MSC examples. I have some limited experience programming it, too, so maybe it will turn out to be easier than the other two.