The big win for cast iron is its higher heat capacity. It takes longer to heat up, but once hot, the temperature drops much less when you add cold / room temperature food to fry, resulting in crisper and less soggy output.
Heat capacity is not a straight win, it's a double edged sword. Like you said temp drops less when food goes in, and that can be great in certain applications. However, you also lose a lot in the way of temperature control. The slow rate at which it heats up and cools down means finer temperature adjustments are much harder to do.
In my experience the searing/frying performance of carbon steel is not an issue for the overwhelming majority of things I cook, ergo I don't really care that cast iron has a higher capacity. The tighter temperature control, on the other hand, is always nice.
I still use my cast irons sometimes for things where that heat capacity is really nice (searing a huge steak or something like that), but I use carbon steel much more frequently.
It also depends on your stove. Mine’s electric and alternates the heating element on and off to try to maintain the desired temp. The heat capacity helps to smooth out temperature fluctuations but if I had a nice gas stove I’d probably switch to carbon steel.
