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 (PMU) is just the control and conversion path that moves energy into and out of the battery. So in a strict hardware sense, you can say the system “runs off the battery” rather than directly off the charging circuit.
But the charger does supply power when connected. 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. Once the battery reaches the optimized charge level, it basically acts like a conductor for the energy and the battery is minimally chemically aging. Heat production also stabilizes because the chemical reaction has reached an equilibrium.
Battery “chemical age” (the real wear factor) is driven by the following, heat, especially sustained high temps above 95°F. Number of full cycles, roughly 1,000 cycles to ~80% capacity under optimal conditions. Time, batteries slowly age even with light use Extreme charge states kept for long periods always at 0% or always at 100% in hot environments.
The act of “charging while using” is not itself a major stressor. The phone’s PMU manages current and voltage, and when the battery is full, charging essentially stops or goes to very low trickle to maintain 100% or optimized charged level.