It's an accumulator with a rubber membrane. Air on one side, water on the other. If water comes out when you depress the pin on the air valve, the membrane is split.
The problem was present before and after I pumped the accumulator up, so I think you're right -- likely unrelated.
The original pump was leaky around the filter, which is why I replaced it. There was a lot of water under the floors! The chipboard underneath the pump was absolutely sodden.
I don't think that's the issue -- I get a decent run time on the taps when the pump is shut off. The pump continues to run some time after the taps are shut off too.