Why does Arduino SD card module only work when multimeter attached?
The probes are likely acting as capacitors and slowing the clock signal or maybe killing off a false edge that your card might see causing it to be read incorrectly.

Try connecting a 100pF cap across SCK to ground to see if this also makes the system work correctly.

If this does not work try 1nF and if it doesn't work with 1nF then the addition of the probe cables may be reshaping the SCK pulses in another way due to reflections and this points to the code not reading the card data at the correct time. 

I'd have thought the clock in to the card would be registered on the +clock edge and maybe data read out on the -clock edge. (This is just a simplified vision in my head of course). Maybe the cable, via extra capacitance or small reflections is shifting the clock slightly allowing your "erroneous" code to work?