TLDR : Are the plates of a battery more like a capacitor with excess charges on the plates?
Yes. This is why a (charged) cell has a non-zero open-circuit (no external circuit) voltage across. Despite the fact that there is no external circuit through which charge can flow, the reactions of the plates with the electrolyte result in one plate having a deficit of electrons (positively charged) and the other plate having an excess of electrons (negatively charged).
Essentially, it is this separation of charge (and the associated electric field) that 'halts' the (net) chemical reactions at the plates and produces the constant open-circuit voltage.
If electrons are allowed to flow through an external circuit (load) from the negative plate to the positive plate, the reactions proceed and the cell discharges.
In the case of a rechargeable cell, an external source can move electrons from the positive plate to the negative plate essentially reversing the chemical reactions at the plates and the cell charges.