Why a cap could cause product failure even though it passes an ESR test could be as simple as the capacitor is breaking down under the applied voltage in circuit, which can in some cases in the hundreds of volts, whereas that is not the case when using any ESR meter.
As to why you are getting conflicting ESR reads with what you describe as significant, I can't hazard a guess. I personally have 3 different ESR meters, but they all agree with each other within 0.1ohms.