A quick look with the scope on the SDIO signals shows that CLK is ~50MHz and D0-D3 are active during transfers, so I think we're in 4 bit mode 2, which is good. The edges are a little too round for my liking, so I may try swapping out my series termination resistors (not in the pic, on the back of the PCB) for smaller values.

I don't have indication from the software end that the SDIO bus is running into bad transactions. I'm not sure if I should expect Linux to complain, or if the drivers have error counters that I can see through sysfs or by turning on some debug flag. Unfortunately all three pieces of the puzzle (RPi SDIO, ESP8266 driver, and my janky SDIO bus) are all in question, so it's hard to tell where the fault might be.

Maybe someone at Espressif wants to open their ESP8089 driver SDK?

EDIT: The signals look decent. My probe's compensation was way out of whack.