What failure do you observe? Or is it just a guess about possible failure?
In which regard do you see lack of "stability"?
In an application with tight frequency stability requirements, load capacitors need to match the value specified for the crystal, otherwise the nominal frequency won't be achieved. Ethernet PHY has relative large frequency tolerance about 50 ppm, respectively varying load capacitance doesn't involve application problems.
Some oscillator circuits have a restricted load capcitance range and might even not oscillate with marginal capacitor values. If the oscillator is critical in this regard (don't expect it for the present device) it's advisable to choose the crystal to fit the device specification.
Maximum drive level can be problem with small crystals and high level oscillation circuits. Increasing the series resistor is a possible option to reduce the drive level.