I figured out what's up with the 5fps over the cheap dongle I have. In fact, I ordered a second, newer looking dongle, and it also gave me only 5fps.

Essentially all of these low-cost HDMI-to-USB dongles are based on the Macro Silicon MS2109 chip. It took some sleuthing, but I eventually found a datasheet for that chip. The chip has two output modes: YUV422 and MJPEG. MJPEG is what we really want, and the chip can output up to 1920x1080@30fps. In YUV422 mode, it's limited to 1920x1080@5fps. The dongles (at least the ones that I have) are configured for YUV422 mode, and that can't be changed without modifying the firmware in EEPROM. I don't know why the manufacturers selected that very limited output format.

Meanwhile, it looks like the go-to chip for the recent generation of those dongles is the Macro Silicon MS2130 (datasheet). In addition to the faster USB 3.0 interface, it supports output of 1920x1080@60fps for both YUV422 and MJPEG outputs. (The datasheet says the chip can be programmed to output 4K@15fps, but I haven't seen that mentioned in any advertisements for these things.

The older MS2109 dongles can be had for around US$10, and the newer MS2130 dongles go for around US$20. For this sort of project, that price difference doesn't matter too much. I got a dongle with an MS2130 chip:

Bus 002 Device 094: ID 345f:2130 UltraSemi USB3 Video

It does, indeed, emit video at FHD@60fps. 

I could wish that it emitted MJPEG, but at least I am seeing the full resolution and frame rate.