The min-pop is not supposed to return anything, yes. We are only interested in the final contents of the heap. So no sorting required. You can eg give the results of the whole operation as a bitmap over the inserts, so you need to return a linear number of bits.

Min-pops and inserts will in generally be interleaved. The prototypical example has blocks of 2 inserts and 1 pop repeated n times. (All other interleaving patterns can be reduced to this one in linear time.)

QuickSelect doesn't work for this.

QuickSelect or Median-of-Medians approaches work if you only have a small constant number of interleaved blocks (of any arbitrary internal length). Like eg all the inserts first then all the min-pops is equivalent to finding the k smallest elements.