How exactly does a battery work (specifically Pb/H2SO4/H2SO4/PbO2 lead acid battery)?
In order for oxidation-reduction chemical reactions to occur, the electrons must somehow go from the anode electrode where the oxidizing (electron donor) takes place to the cathode electrode where the reducing (electron acceptance) occurs.

Ideally, the electrolyte, in this case sulfuric acid, should allow the movement of positively charged ions but prevent the movement of electrons through it.  This forces the electrons to go through a circuit, external to the cell, to go from anode to cathode thus enabling the redox to occur.  Without electron flow, said redox reaction will not occur.

In the real world, there will be some leakage current through the electrolyte and other parts of battery.  We can eliminate the leakage with a more insulating electrolyte, but this would hurt the charging process and make the battery less rechargeable.  So, what we have is an economical compromise.