Large cogeneration sites (where they use the waste heat from electrical generation for process steam, building/district heating, etc.) actually do run in grid-backup mode. An example is MIT's cogeneration plant (a couple of big natural gas turbines on Vassar street) -- a lot of universities do this since they can use the steam for heating, and a lot of industrial sites do it for process.
It comes down to cost and zoning/permitting. It's much easier to get a permit to run a generator for backup use than to run one 24x7. It's also hard to get a 1-10MW plant which is per-KWh as efficient/inexpensive as the grid (although now that natural gas is about 20% of what it was when I last bought it, gas turbines actually are cheaper than industrial tariff grid power, if you have good gas access...). Being able to actually use the waste heat is what makes the combined cycle efficiency worth it.
There was a crazy plan to run a datacenter on a barge tethered to the SF waterfront, for a variety of reasons, but a primary one being power -- the SF city government wouldn't be able to regulate the engines/generators on a ship running 24x7.
My university had a big cogen plant, but it was never designed to power the entire campus (it was only able to do so at around 3 AM). Aside from providing heating and power, because it was run off of natural gas it qualifies for clean energy credits, which the university makes money off of by selling on the market.