The inefficient use of generators means that a great deal of diesel is being burned unnecessarily. These emissions contribute to climate change and poor air quality in cities

These downsides are noteworthy, but I think the people/organizations who are employing them at their events/sites have already made the decision that these aren't factors, and rather they are focused (reasonably so) on reliability and price. The big pivot point in energy storage will come in the not too distant future when battery tech turns the corner and will be seen simply as the more effective option. 

"...batteries can stand in for the grid during downtime..."

No Ben, batteries cannot "stand in for the grid," and never will.

Batteries can shave some of the peaks in demand from diesel generators, which stand in for small areas of the distribution grid - if lightning strikes a power pole in a neighborhood, for example.

Another attempt to greenwash both diesel fuel, and the natural gas being used to charge grid-scale batteries (they don't charge themselves).

I think this is pure theoritical assumption trying to magnify storage battery capability. Actually ther is hibred  diesel generator set with renwable and backed up with battery. Have you justify it monewise? Thanks in advance.