The most important characteristics of a capacitor are those indicated on the capacitor, that's:
...
That would be quite an oversimplification.
Afaik, the markings on a cap are mainly for identifying it, not for characterizing it, although in many cases the markings are conveniently obvious and enough for the latter, too. What really is important is defined by the circuit the cap is in (or schematics etc.).
E.g. some caps have no markings on them at all, would that mean no parameter is important?
I do not know of any electrolytic cap to have ESR marked on them, but in many circuits (especially in various power supply types) the ESR can be more important than the capacitance (as in, the allowable range of values can be tighter), even so that ESR must be within some range, instead of being less than some value, otherwise the circuit won't be stable. (Imho, a stupid design idea, but I guess the goal is to save 0.1 cents per device, ignoring any future costs of more difficult repairs or even full replacement of the device).
Also, consider e.g. ceramic caps, where some types have a huge reduction in effective capacitance as one gets higher and higher in the voltage, and the nominal capacitance is stated at some low voltage (e.g. TDK rates capacitance of ceramics even down at 0.5+/-0.2Vrms, another at 0V+/-?), instead of at rated voltage. If one accidentally replaces C0G with a X5R or such types in a circuit that works at higher voltage, things can fail to work. (I'd have given an example for the reduction, but of course, now that I want to find such graph, I can not spot any in the datasheets, and just couple weeks ago I seemed to see them in every single one...
)
Of course, if the capacitance is out of spec already at a low voltage, it will be bad cap, but hey it was the same for the leakage current. According to you, test has to reveal all the issues, even the high voltage ones, or it is of 100% unusable. Right?