I guess we all may agree to following points:
+ the CMOS IC is designed for crystal use, even at 3v3 according to NXP spec
+ it works with the 8 MHz crystal (at least badly/sometimes) -> the IC is at least not fully damaged
+ it does not work with the 7.3728 MHz crystal (data is available)
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My 1 step of analyis:
+ in principle it is a feedback (looped) amplifier
+ a good designed oscillator is a badly compensated feedback amplifier, it thus start to oscillates
(at amplification you don't want it, in this case it is required, you need it)
1st conclusion = it does not start (or stops again) the oscillation, because the compensation is wrong (in our case: too much damping)
My 2nd step of analyis:
+ too much damping means a too low ac load - a too low resistance at the target frequency of 7.3728 MHz
+ the damping circuit is build out of the crystal (a band path at target frequency), the series & feedback resistor and the clamping capacitors.
+ it ~works at 8 MHz, but does not at 7.3728 MHz (frequencies are very close to each other) with the same rest of parts in the compensation
2nd conclusion = the ac specs of the both crystal must be significant different
(I couldn't find an info, if the data sheet is for both crystals, but I don't believe)
MY SOLUTION
= either increase the amplification (reduce the effect of the damping)
--> reduce the parallel resistor (typicall one starts with 10 MOHm for CMOS, here we have 3V3, thus I would start with 1 MOhm)
= or reduce AC load (increase the ac resistance) of the damping circuit
--> you may reduce the resistor in line with the crystal (or directly start with zero Ohm)
--> you may reduce the clamping capacitors/increase there impedance (or start by leaving them away)
You may also calculate the optimum AC load using the equivalent circuit of the crystals and taking the values out of the data sheet

Does the comunity agee to my analysis/conclusion/steps to solution?