Purpose of electrolyte in a galvanic cell
Unless I'm badly misunderstanding your question, I think your confusion arises from an error in the balanced chemical equation you wrote.  As you have it, a salt is reacting with a strong acid to make another strong acid and a new salt. This reaction may occur spontaneously as written (I haven't figured its Gibb's free energy) , but it's not the oxidation-reduction reaction that drives a galvanic cell.  In fact, it's not an oxidation reduction at all, it's some sort of acid-base reaction, and all of the reactants and products remain in solution anyway.

The actual chemistry that occurs in this system is between the zinc anode and the acid.  

$\ce{Zn + 2 HCl -&gt; ZnCl2 + H2}$

From this it is much clearer why the net ionic equation is as you wrote.  The zinc sulfate could be replaced with zinc chloride with almost no effect on the battery's potential.  In fact, the zinc solution exists only to create standard conditions for the cell and to carry electrical current through the solution - if it were replaced by a nonreactive electrolyte solution the cell would still function, albeit at a different (higher) potential.

You might then ask what the purpose of the hydrogen cathode is.  Again, it's to create standard state.  If you did this in reality, you would be using some sort of inert electrode at the hydrogen end to carry the electrical current into the acid solution and likely would just vent the hydrogen to the surroundings.  

In short, this galvanic cell does not function because of a reaction between its electrolytes, nor because of a reaction between its electrodes.  Instead it is ultimately a reaction between the anode and the cathode's electrolyte.  The anode's electrolyte and cathode are required for electrical continuity and are used in examples of this sort to put the cell in standard state.  Beyond that they don't serve much of a purpose.