The transmission has a factory thermostatic bypass valve that purposely restricts flow to the cooler until the fluid reaches about 210°F. An aftermarket deep pan (like a Mag-Hytec or PPE) will add extra fluid capacity—which means it takes longer to heat up—but that factory thermostat will still force the transmission to reach its 210°+ baseline.
If you actually want to see lower temperatures, the real fix is a Thermal Bypass Valve (companies like PPE make them). This replaces the factory thermostat to allow full-time flow to the transmission cooler right from the start, which can drop your baseline operating temps by up to 35°F. A deep pan is still an awesome upgrade for the extra fluid capacity, cooling fins, and the drain plug, but the bypass valve is what actually moves the needle on temperature.
Transmission Coolant Thermal Bypass Valve, 2020-2026 Ford Super Duty 10R100/10R140, Full-Flow Trans Fluid Coolant Valve eliminates the factory restriction
ppepower.com