In level flight, there's no pressure differential in the VSI, so I
don't see how a leak could matter.

Actually, that's not right. The pressure inside the fuselage is
generally not equal to the outside static pressure. In general, since
there are openings, the pressure inside the fuselage is actually lower
than static pressure, due to venturi effect. That's why IFR airplanes
(usually) have static ports, rather than venting into the cabin, and
why the use of an emergency static source (which vents into the cabin)
causes altimeter and airspeed errors.

In a descent, the pressure in the static system is higher than the
pressure in the VSI case, so there is a continuous flow of air from the
static system into the VSI. The VSI reads this flow of air as a
descent.

In level flight, the pressure in the VSI case is the same as the
pressure in the static system (it has equilibrated) and thus a little
higher than the pressure outside the VSI case. If there is a case
leak, the air leaks out of the case. This means replacement air comes
in from the static system - just like in a descent - and the VSI
indicates a descent.

Michael