If it had been maybe 50 to 200 birds, I would believe any story they told. Birds are insensible creatures and startle easily.
However, 5000 birds is a bit much of a sample population, even among millions of birds, to have all been of the same species. There were no others of any other species among the dead.
My theory is a sickness, either biological from nature or engineered by man. It could very well be a small test of chemical weapon that is designed to kill birds so as to spread disease in a area. It could be a very useful, non-invasive way to eliminate or neutralize a population.
It's just too sudden to be a naturally occurring sickness, and too many for it to have been fireworks. Birds die by flying to buildings (an obvious cause of death that doesn't simply vanish) and those that are startled by fireworks do fall out of trees. However, it is not common for this to happen in numbers outside the low hundreds.
The dead birds of the same species in Louisiana dropped out of the sky onto the highway. There is no cause for that, and as far as I know, no solid explanation has been offered.