Note that for the purposes of this question, I am only interested in the scenario described below. By 'steady-state' I mean to exclude from consideration the short-term effects of changing the flap setting during a turn.
In the YouTube video Should I Use Flaps in a Turn?, beginning at 10:59, there is a claim that, at a given airspeed and bank angle, having the flaps lowered will decrease the turn radius. This is followed by a demonstration in a PA-28 where airspeed, bank angle and height are being held as constant as possible (70 kts, 30 degrees and 2,000 feet respectively) through two 360-degree turns: firstly with flaps up and secondly with 25 degrees of flap. It looks pretty convincing if we assume the bank angle and speed were adequately matched in both cases (though there are some variations apparent in those shots showing the panel and forward view during these turns).
The explanation given is this: "The turn radius of an aircraft is directly related to the horizontal component of lift which, as you may remember, is how we get an aircraft to turn. Well, what do flaps do? Yes, they increase our drag but they also increase the amount of lift that our wings produce, so instead of increasing our bank angle (which increases our load factor) we can just lower more flaps. This allows us to tighten those turns up."
The doubt I have over this explanation is that, while (at least in the normal flight regime) lowering the flaps increases the lift at a given airspeed, air density and angle of attack, what's being held constant throughout these tests are airspeed, air density, and the requirement for the vertical component of the lift to be equal to the weight (this last constraint is on account of these tests being conducted at constant altitude, so there is no vertical acceleration).
Given that both tests are performed at the same bank angle, however, this last constraint on the vertical component of lift means that the total lift would be the same in each test, and the same can be said for its horizontal component. It is the latter which acts as the centripetal force for the turn, and the only other variable in determining the radius of the turn is the speed, but that is also the same in both tests.
If this phenomenon is real, what am I missing or getting wrong in this analysis, and how do we calculate or estimate the size of the effect?