Not to mention the horrific peak power draw. It took people a while to figure out that the things need a fair bit of close-by capacitance on the power rail or they crash.
For example, put a sleep(100us) as a hook before packet transmission to allow capacitors to recharge between packets.
Had to do this on a design powered by a cr2032 because the peak power draw from those batteries is really limited.
Though note that for most of its life, the cr2032 will deliver slightly below the ESP's minimum spec of 3V. From experience, that's not really an issue.
If you can get away with spending most of your time in the ESP's deep sleep state however, battery life is gonna be way better, and that's probably what you'd want to do if you're using a cr2032. In deep sleep, the ESP32-C3's data sheet says it consumes around 5µA (0.005mA). With 200mAh, this gives a battery life of 40 000 hours, or 4.5 years. At those time scales, battery self-discharge becomes relevant, so let's say a couple of years.
So a decent mental model is: you can do a few hours of cumulative compute over the course of a couple of years.
Unless you decide to boost the voltage of the cr2032 to be within the ESP's spec. In that case, the whole deep sleep discussion might be moot; I suspect the regulator's own power draw would absolutely dominate the 5µA of the ESP. But I'm not super familiar with the world of power regulators, maybe there are ultra low power regulators which can do the job with minimal losses even in the µA regime.
I also don't think it's too unreasonable to use a C3 as an MCU in settings where a radio isn't required. The IC itself (aka not the wroom module etc) isn't that much more expensive than equivalently specced MCUs without a radio, and if you're already more familiar with developing for the ESP software ecosystem it might well be worth the few pennies extra. The ESP32-C3FH4X (so the IC with 4MB onboard flash, super easy to integrate into in a design) only costs $1.6 per unit on LCSC for 100 units (and mouser etc is similar).
This is exactly a thing the ESP32 with WiFi cannot do if powered by a standard alkaline battery. The battery would drain in a couple of weeks.
And on top of that, it probably doesn't work at all, as it cannot even send once because the peak power draw is too high for an alkaline battery.