Don't mind the gallery, POA tends to be snarky for sport. In non-swept, subsonic wing planforms of moderate to high aspect ratios (essentially these light piston clam traps), flaps change the effective camber of the wing, which shift the Pr and Tr curves up via extra drag (combined). This has the effect of hurting the angle of climb capability of the airplane, lowers the service ceiling as well. But it also changes the VX and VY numbers as provided by the OEM (to the left), because the calculation of Vx and Vy is made with the wings in the clean configuration, and labeled for max gross weight.
Conversely, decreased weight for the same powerplant and wing configuration increases the power loading, which increases the angle of climb capability (excess thrust and power points in the curves respectively). That's why naturally, the airplane performs better in the climb when it's lighter.
But that's climb. Now let's talk about your question, takeoff. There is a competing relationship between 1) acceleration to liftoff speed, and 2) liftoff speed calculation as a function of flap angle. Piper essentially is telling you that two notches of flaps, eg 25* ,is the optimum combination for minimizing ground run. Why is that? Because the effective camber of the wing with flaps at 25* yields the quickest convergence point to the attainment of liftoff speed, which minimizes ground run (all weights) when compared to flaps 40*, flaps 10* or flaps 0* takeoff attempts.
You have to remember from the prior discussion, that flap angle changes camber, which changes Cl per alpha, and that changes liftoff speed. Yes, flaps 0* will have less drag during acceleration than flaps 25*, but the aircraft has longer to attain the higher no-flap liftoff speed. Conversely, 40* has the slowest of all liftoff speeds, but the drag penalty in acceleration still yields a ground run longer to reach this lower liftoff speed. Piper did the homework for you and showed you that two notches, not one, not three, not zero; 25* gets you the minimum ground roll to wheels off the pavement.
Flaps 10* use in your clubs case? Sorry, it's just placebo. You'll have a marginally shorter ground run than flaps zero, but your climb gradient will be poorer than the clean wing takeoff counter. If you want to minimize ground run, 25* is the optimum choice. If you're worried about settling in high temp/DA days and prefer to have that "positive feel" when climbing away,
flaps zero and a longer ground run to liftoff speed closer to Vx and Vy is the ticket. Flaps 10 is generally an instrument approach sight picture enhancing setting prior to breaking out the clouds and committing to flaps 25 or 40 on the visual landing. Perfectly fine to use it on takeoff, but you're not gaining anything deemed important when taking off. Flaps 0 or 25 are the answer for the PA-(2,3)X-xxx series.
As to the POH. What
@Ravioli was hinting in his matter-o-fact way is that the POH is a legal instrument for the FAA to hang you with. Citing deviations from the POH, even if immaterial to the accident, is in his opinion is opening yourself for scrutiny and fault-assigning. Bear in mind, POHs have been laden with safety-detracting directives and information. First two that come to mind are the recommendation to lean to 50ROP above 65% of pre-restart Pipers and Cessnas, the other one being non-altitude compensated power tables for a blown conti 360 in the turbo arrow. That one can get real scary real quick, especially with an intercooler mod and a neophyte at the helm.
It's a bit of hyperbole that the FAA is gonna yank your ticket because you chose to fly flaps 10 and porked a takeoff, but the principle behind using the POH as the catch all to cover yourself from the hall monitor types at the FSDO is probably sincere advice.
