But the only ship that can carry it is the Zumwalt, there are only three of them (so the price per gun or round will be enormous), they have very little capability against air targets (because this was deleted from the Zumwalt in a previous cost-saving measure).

So a rail gun on the Zumwalt would basically do the same thing as the AGS. It would maybe have longer range, depending on how many megajoules they can make the gun output, this is still an open problem. And the magazine would have space for twice as many rounds, since they don't need to store the propellant. Given that the Navy apparently just decided that they don't even want the AGS, I don't think they care at all about the capabilities they'd get by installing railguns on the Zumwalts.

They may still install it as basically a pure research project, but if money is tight...

The AA capability of the railgun was "deleted"? Where did you hear that?

No, of the Zumwalt destroyer. Originally it was to have a "dual band" radar (an S-band search radar and an X-band tracking radar) and the capability to fire Standard Missile-2, -3, and -6. Eventually, the S-band radar was deleted and the integration work to handle the Standard Missile was not done. When the Navy cancelled the Zumwalt orders to instead produce more Arleigh Burke destroyers, Raytheon offered to do additional work to integrate the Standard Missile, but the Navy was not interested, so now the only AA weapons it carries are Sea Sparrow missiles for self-defense. This is all from Wikipedia.

I can't imagine them not doing that integration work if they add the railgun. Even if we assume it's just a technology demonstrator, they'll want to know how well it works against missiles.

OTOH, only the third one is getting the railgun if it happens at all.