They only have the ability to deal with one emergency at a time? That strikes me as sub-optimal, especially if the "emergency" is contained within a single parked aircraft. Diversion is probably costing all the airlines involved a ton. I think it would be worthwhile for emergency facilities to have a backup.
Anyway, I suspect the real reason is that an emergency is a dynamic situation, and an airport is a large complex barely-controlled environment. Sure, probably parts of the airport could stay in use, but it would take a lot of coordination and analysis to do so safely, and there's really no time to do that. So you just shut it down.
You know what's going on now, so it's easy to say; but if you're running the airport and all you know is "aircraft on fire", what do you do?
As for the cost, yes, but airlines are fairly risk adverse creatures and I'd imagine they'd have adequate insurance to cover such eventualities.