I've heard the same thing in a number of places and accepted it as gospel. But I just did the calculation for a PA-28 with a wingspan of 10.66 meters, doing a standard rate turn at 100 knots. I found that the radius of such a turn is 955 meters, so the speed difference between the wingtips is just a hair over 1.1%. Lift is not generated by the wingtips alone but by the whole wing so measured over the whole wing the speed difference, and therefore the lift difference, is more like 0.5%. I can't believe this is really responsible for the steepening of the turn.
That's part of the equation, but the velocity is only part. While there may not be a huge airspeed difference between the outer wingtip and inner wingtip in a turn, there's a difference in the angle of attack. Additionally, the difference can be magnified in a descending or ascending turn, and is further altered when the airplane is not flown in a "coordinated" ball-in-the-center condition.