Could a lead-acid battery electrolyte be replaced by hydrochloric or nitric acid?
In the lead battery it is very important that the electrodes transform into insoluble lead sulfate when discharged, especially at the cathode where lead(IV) oxide is turning into lead(II) sulfate. Consider the two half reactions of the lead-acid battery:

$$\ce{Pb(s) + HSO4−(aq) → \color{red}{PbSO4(s)} + H+ (aq) + 2e−}$$
$$\ce{\color{red}{PbO2(s)} + HSO4−(aq) + 3H+ (aq) + 2e− → \color{red}{PbSO4(s)} + 2H2O(l)}$$

Lead nitrate is soluble, so metal ions would go into solution. The lead(IV) oxide would not form again from these ions on charging, metallic lead from the electrode would dissolve. The battery would no longer be rechargeable.

Lead chloride is insoluble, but lead(IV) oxide on the cathode would oxidize chloride into chlorine. Thus no more battery.