It’s true that the iPhone’s main power rail for the system (CPU, display, radios, etc.) is tied to the battery. The charging circuitry is just the control and conversion path that moves energy into and out of the battery. So in a hardware sense, you can say the system “runs off the battery” rather than directly off the charging circuit.
Here’s how it works, as explained to me by an Apple Store Lead Genius with about 20 years experience.
The iPhone will run primarily off the wall charger. It will fast charge to 80% (generating a lot of heat) and stop charging if the iPhone gets too warm or it reaches it’s optimized charging limit. At that point the device has been primarily running off the charger but supplementing with the battery, if needed, during high energy demand periods.
Slow Charging may resume to 100% if the device cools enough to support continued charging. If you have optimized charging enabled, charging will resume so device is at 100% near the regular wake up time.
When the iPhone is connected to a charger the external power supply is the primary source of power. The phone draws as much as it needs from the external supply. Any extra goes to charging the battery. If the external supply can’t meet demand, the battery supplements.
So the phone does not simply ignore the charger and run 100% off the battery while plugged in. The charger is supplying power to both the device and the battery; the battery is just in the path, storing and releasing energy as needed.
This type of usage does not in itself cause damage to the battery. Any damage would be caused by excessive heat. I’d recommend getting an iPhone cooler that attaches via MagSafe or a clamp and has a Peltier element for active cooling, if you’re making a regular habit of this.