STM32F103C8 SPI1 clock frequency too low (~32kHz)
If I start the Saleae Logic with sample rate 16 MS/s, I get a message
box 'we're sorry, but the device was not able to keep up with this
sample rate ...', and no data captured.
Lowering the sample rate down to 500 kS/s results in data capture, but
I can see that the clock frequency is about 32 kHz, not ~8MHz as
expected. Changing BR (bits 5:3 in SPI_CR1) brings nothing.

To capture an 8 MHz square wave, you're going to need a capture rate of at least 16 Msps, preferably higher.
Trying to capture an 8 MHz signal with 500 ksps simply isn't a viable strategy.
Try again with 16ksps and lower the capture duration to fit within the logic analyzer's memory.  Consider using Logic 2.x software instead to see if it'll work better.