Why does frying in batches actually matter?

Because adding too much food at once drops the pan temperature, and below a certain point the food steams instead of browning — producing grey, soggy results from the same ingredients and the same pan.

The mechanism. A hot pan holds a finite amount of stored heat. Cold, wet food absorbs that heat rapidly. Put in a small amount and the pan recovers quickly; put in a lot and the temperature falls sharply and stays down, because the burner cannot resupply heat as fast as the food removes it.

Why that ruins browning. Two things must happen for a good crust:

The surface must reach around 140°C or above for the Maillard reaction to proceed at a useful rate. Below that it happens extremely slowly.

Surface water must evaporate first. While liquid water is present, the surface cannot exceed 100°C, because the energy goes into boiling it off rather than raising the temperature. Only once the surface is dry can it climb into browning territory.

Overcrowding defeats both. The pan is cooler, so evaporation is slower, and the released moisture has nowhere to escape — it fills the pan as steam, keeping the surface wet. The food sits in its own moisture at around 100°C and stews. This is why crowded mushrooms release liquid and go grey while a few in a hot pan brown beautifully.

What to do:

Cook in batches, keeping earlier batches warm. It is faster overall than trying to cook everything at once and then waiting for it to brown, which it never does.

Use a bigger pan, or two.

Dry the food thoroughly before it goes in — patting meat dry is the single cheapest improvement available.

Preheat properly, and let the pan recover between batches.

Use a heavy pan. Thermal mass is precisely what resists temperature drop, which is why cast iron and thick steel outperform thin pans for searing regardless of the material's conductivity.

Leave it alone. Moving food constantly prevents sustained contact and slows browning.

Salt later for vegetables, since salting draws out moisture.