Here are the main problems with fitness testing (or any kind of testing for that matter): First, you can't test everything. The APFT tests for an extremely narrow definition of fitness (Cooper test is even worse in this regard), heavily skewed toward endurance...
I think that is all true. So perhaps the best way to quantify fitness is by how well you can do what you need/want to. Competition is also a reasonable way.
The problem is also not just measurement. It is generally not even possible to be optimally fit in all categories. For example, if you want strength you need to develop thick muscle fibers. Endurance requires more, thinner fibers so that the blood can reach permeate them continuously to supply oxygen.
If your goal is to do an Ironman then you would measure your fitness in an entirely different way than if your goal was to throw the most beer kegs over a wall in a given time.
I say...set goals and measure your performance against the goals. I personally do this by competing i triathlons. I have no chance of winning but I get an objective measure of my overall fitness every time. That is of course skewed toward endurance but so what. It's what matters to me personally. I also measure VO2max, resting heart rate, and strength in various ways periodically.