It doesn't bind CO2 from air during setting- that's the disconnect.

By converting the carbonate to oxide, the calcium in cement is free to bond with silicates (and to a lesser extent other compounds).

That CO2 released during manufacture is now looking for a new home, which could either be a plant, or a weathering deposit of limestone which could use the CO2 in the air to convert its own carbonate to stable bicarbonate.

At some point it must be cheaper to just cut a massive piece of rock and ship it to the construction site, isn't it?

Depends. Are you taking into account reconfiguring bridges and overpasses to make the loads fit, or factoring traffic management costs into the equation? The truckloads of concrete to make a structure individually take up a lot less space and weigh less than the final structure itself.

I've wondered about that of late. Assuming that fitting stone work was is a skilled craft. But with modern CNC technologies might be worth a look again.

The “separate carbon from stone” step is not 100% efficient