Pressure is a misnomer I expect, the pressure is dictated by the temperature. As the propane is in liquid form the tank pressure is equal to the vapour pressure of the propane at it's current temperature.
In operation the propane tank needs to take energy (heat) from the environment to allow the propane to boil into gas and feed the appliance. If insufficient heat can flow to the cylinder then the propane temperature will drop and the pressure will also drop, reducing the maximum supply rate. The benefit of a larger tank is that the amount of propane in the tank is greater and it will take longer to cooldown during use.
Possibly the supplier has found that people using smaller tanks have seeing performance issues, they may or may not understood the fairly complicated thermodynamics and trying to explain these to the general public they've simplified it to saying it's a pressure issue.