Leclairage:
I really do not want to get into a huge technical discussion about Performance "A" requirements since such aircraft as the PA-28 are probably in the Performance "X" (no performance) category.
However, you are just about right in your assumptions.
Imagine if you will taking off from an enormously long runway and then finding that you simply cannot climb. All will be well until you meet or try to avoid the first obstacle.
If your weight is too high or the temperature is too high or the airfield is too high then you WILL have a problem.