Utility installations have much cheaper solutions for cooling, the balancing is much easier with lots of parallel cells, and even the chemistry benefits from smaller charge/discharge currents.

We're not down to $30/kWh yet, but we're not that far off. Certainly well below $300.

Yes, there are two things that probably contribute here.

One is (as you guessed initially) "storing only a small percentage of the total energy output and giving a diluted price with the partial storage cost spread over the total output." The majority of the solar generation can be consumed immediately without taking a trip through the battery. The battery capacity is needed to serve peak demand periods in the late afternoon/early evening, which last just a few hours.

The other is that you are probably lowballing the number of charge/discharge cycles at 2190. Unlike EV batteries, stationary batteries for grid demand don't ever need to "supercharge" at high rates or rapidly discharge for high acceleration. Careful moderation of charge/discharge rates can give double or more lifetime compared to identical batteries that undergo fast charging and discharging.