Given that hydrogen fuel cells can be used to run cars, why not use them to run power stations?

Though I'm not overly familiar with the physics of it, hydrogen fuel cells operate by exploiting the energy released when hydrogen (H2) and oxygen (O2) recombine (in a 2:1 ratio) to form water (H2O).

I'm assuming that this process is energetically beneficial; that is, you produce more energy in a fuel cell than it takes to produce the fuel.

I'm also assuming that the only thing stopping widespread deployment of hydrogen fuel cells in cars (which I think is the venue most often discussed for fuel cells' use) is the logistical problem of refueling such cars in everyday use: containment suits are needed when pumping the very-cold liquid hydrogen—or is it oxygen? or both? Anyway, a very-cold liquid—into the cars. And, while petrol stations are conveniently ubiquitous (for the most part, anyway), none of them stock hydrogen (and/or oxygen—whatever we decided was the stuff being pumped); this is largely because there are no hydrogen fuel-cell–powered cars in everyday use. It's the proverbial chicken-and-egg situation (let's pretend, for rhetorical purposes, that that one wasn't resolved (it was the egg (no, I'm serious))).

These problems don't, however, stop hydrogen fuel cells (or the same technology, perhaps not in an actual cell) being deployed in centralised systems—like power stations. The chicken-and-egg problem is resolved because popular demand isn't needed to make it commercially viable to supply the technology, the infrastructure or the fuel itself.

It's definitely possible (at least I'd be very, very, very surprised and bemused if it weren't) to run a power station using hydrogen fuel cell technology: most (perhaps all) modern power stations operate by heating water. Coal-fired and nuclear fission-powered power stations both use the energy released from their respective reactions to heat water, which becomes steam; as the steam rises it turns a turbine; this turns a magnet in a solenoid (a tube-shaped coil of wire), which induces an electric current in the solenoid.

Hydrogen fuel cell–powered cars have wheels that turn. Attach a magnet to one, put it inside a solenoid et voilà.

The benefits of using hydrogen power, rather than fossil fuels or nuclear fission, to produce electricity would be plenty: removing the “kettle” stage from a power station would be a novelty; it would probably increase efficiency lots-fold, and it certainly (defined like “definitely” above) wouldn't decrease efficiency.

The environmental benefits are pretty substantial too: hydrogen and oxygen exist in abundance in water, and the only by-product from the fuel cell reaction is hot water. As long as this hot water isn't pumped into rivers at such a rate that a river's temperature rises significantly, which would piss a lot of aquatic animals off rather a lot, to say the least, the environmental concerns are nil—you can safely pump water into the atmosphere (almost) with abandon.

There must be an obvious stumbling block that I've overlooked; otherwise, why aren't all the cool More-Economically-Developed Countries doing it? (And I doubt it's collusion with the fossil fuel industry—there are lots of rich countries without a vested interest there.)