The insulation in the bag is irrelevant. It's not comparable to a person being in a bag. With a person, there is a bunch of metabolism trying to warm up the air inside the bag. This will work, and keep a person much warmer when protected by a bag. With the footballs, there is nothing adding heat to the inside of the bag. No heater, no metabolism. The air in the bags will drop to ambient air temp.
The rain water plays an interesting role. Since it fell from altitude, it is likely colder than the 48°F - 50°F reported near the end of the 1st half, which is a "dry bulb" temp (i.e., "thermometer under an umbrella"). But the water's primary role is as a superb heat conductor / heat sink. A couple hundred times more effective than air conduction. This would have brought the temp of the air inside the ball down much faster than bringing them inside & drying them off would have brought the temp back up to room temp.
In the experiments that I ran, after dropping them to 50°F with a cold bath & cold air, the pressure inside the balls was STILL significantly low three full hours after I dried them off & sat them in a 71°F room. The moisture in the leather of the ball, the dependence on pure conduction and the lack of convection keeps the temp of the air inside the ball low for a long time. And that temperature is all that matters to the internal pressure.
It should also be noted that having cold air on the "measured" (i.e., "inside the ball") side of the pressure gauge's bellows & warm air on the "atmospheric reference" side WILL produce a gauge pressure reading that is too low. The assumption in the gauge's calibration is that the temp of the air on both sides is the same.
The size of this error depends on the details of the gauge's design. It'd have to be determined experimentally.