VPI
YOu don't climb better at higher speeds because you are producing more lift. Go back to your diagram that shows the forces acting on the acft in a climb, or S&L flight. Lift will always = weight and has little to do with the rate of climb at a steady state. You will notice that the faster you fly the lower the attitude will be which means that you have a lower AOA, AOA directly affects Co efficient of lift which if you multply that by your, 1/2 rho V Squared S, part of your lift formula then L = W. Increas V decrease AoA. In a climb the Trust and drag lines are inclined so some thrust will carrying some of the weight of the acft. In other words it doesn't matter how fast you are flying, in a steady state L=W and T=D. In a climb the excess thrust = ROC.
Enjoy your time on teh P3, I only managed a few years on it but thoroughly enjoyed every minute of it. When people go on about real acft they are talking about the P3.