For one you can get the Mayer-Vietoris sequence in homology almost immediately from a homotopy pushout, but depending on what angle you look at it from, that may not be that useful (as in, to prove the theorem, you need to define the functor $H_*$ anyway, and then you probably already know it satisfies M-V).
You can deduce a group structure on the Eilenberg-Mac Lane spaces, see this previous M.SE answer. (In fact it is probably possible to prove the existence of the Eilenberg-Mac Lane space using Dold-Thom)
At a philosophical level I can do no more than point you to Thomas Barnet-Lamb's excellent piece on Dold-Thom, and some of the reasons why it is important.