Connecting Saleae Logic 8 to MISO (pin 12) on Uno 3 w/ NRF24L01+ halts program
It would appear from your description that the leads on your logic analyzer distort of the SPI MISO waveform and the Arduino does not read incoming data correctly. 

To avoid this, you could place an digital isolator between the Arduino and the Logic 8.  

Any decent oscilloscope should also allow you to read the SPI bus.