STM32 SDIO works only while probing clock signal
Reducing the clock signal pin's drive strength to GPIO_SPEED_FREQ_LOW as per the suggestion from @mkeith dramatically improved the signal:



(nevermind the probe compensation being a little off, tweaked it after the fact)

This allowed me to change the clock divider for the SDIO peripheral to 8, improving the performance of the card. Before changing the drive strength, I had it at 24 to get anything working. And most importantly: I can now access the SD card without probing the clock signal ;)