Why doesn't the Clock in Hill Valley work in BTTF2?

Quite simply, it was never fixed.

Lightning from an electrical storm struck the lightning rod above the clock tower and travelled down a cable which was wrapped around the metal hands on the clock face at precisely 10:04pm on November 12, 1955. The sudden jolt of electricity damaged the clock mechanism and it has never worked since. The Hill Valley Preservation Society kept it in this condition as part of the town's history and heritage.

No the lightning rod did not stop the freeze of the clock at 22:04:00. The rod just made the lightning go in the flux capacitor. It froze and nobody had a plan to fix the clock. Everybody planned to keep the clock the way it is as if its just a fake clock just for Hill Valleys history and heritage. The lightning went through the flux capacitor after it hit the clock.

If the clock had been fixed, it would greatly diminish the significance of the story, as though Marty and Doc's activities had no lasting effect.

Of course, a real community with such clock tower would naturally go to some trouble to fix it, probably even replacing the (presumably) melted mechanism. Otherwise such neglect would tend to attract vandalism and decrease morale generally.

(afterthought)

Furthermore, the civic minded lady in 1985 is soliciting support to fix the clock, but no one seemed interested—when she shakes her coin box, it sounds like only 50 cents is in there.

This is a good question, and to my knowledge it's never answered on-screen.

But what's interesting is that in order to prevent a paradox there must be an answer. The clock needs to be broken (or, at least, everyone needs to think it is broken).

If the clock works, then there's no need to fix it, which means no fliers being passed out by the Hill Valley Historical Preservation Society. No fliers means Marty doesn't have one to show Doc in 1955. But even more fundamentally, if the people of the city don't believe the clock was struck by lightning at 10:04 PM on November 5th 1955, then Doc and Marty have no way of anticipating that lightning will strike then. That means Marty can't get home. Marty not going home means lighting will strike the clock and break it, which allows for the flier and all that, which means they will know when the lightning strikes, which means they use it, which means the clock doesn't break....paradox.