A fundamental question about crossover distortion
Am I mistaken that crossover distortion only happens when using single
  supply?


Yes you are mistaken.


  Or is my simulation wrong since it doesn't reveal any sort of
  distortion at the output?


You need to add a load to the output to be able to recognize cross over distortion. Make sure your input voltage is several volts p-p too.

@Colin spotted the other mistake - Q2 is connected incorrectly.


It will work somewhat with C and E reversed but not as well.