Why polymers have a range over which they melt instead of having a specific melting temperature?
No. Although we need to be careful in defining what is and isn't a "polymer". The nature of (nearly all) polymers is that they are (covalently) bonded (in at least one dimension) over large (relative to atomic diameters) distances. The nature of a liquid is that is exhibits no long range order, no long range "connectivity". There are, as far as I know, no polymer crystals, in their most ordered configurations they are quasi- or semi-crystalline, having disordered zones as well as often but not always having zones of some regularity/order. Often the degree of crystallinity of a polymer is determined by its processing history (temperature, pressure, and stresses applied to it), but even the most crystalline of polymers is not fully crystalline. What this means is that the transition between solid and liquid phases is not sharp; the disengagement of the polymer chains from one another takes place gradually with no sharp/distinct difference between the solid phase and the liquid phase. Perhaps a better way of saying that is that the polymer becomes more and more amorphous but continues to have some long-range order as each chain's motion relative to its neighbors increases. see https://en.wikipedia.org/wiki/Reptation and https://en.wikipedia.org/wiki/Amorphous_solid. Note that the presence of polymer  chains (assuming linear or nearly linear polymers) of varying MW increases the gradual transition from amorphous solid to polymer melt. If the amount of oligiomer is large enough, it may act as a solvent for the larger chains. Note also that the theory of glass to liquid and glass to solid phase transitions is an active area (not yet "settled science").