A bit of a problem with my oscillator
If you have 2 working circuits out of 5, it means that you have either post-soldering contamination, or marginality in the circuit. Solder residue from some no-clean fluxes is hydrophobic, and can be very conductive.

If you really clean and dry/bake the boards, and they still don't work, you might have a marginal component, likely the crystal that is on borderline of oscillator specifications. It could have too high series resistance, or the load capacitance is too high.

There are some simple techniques to check if the circuit is marginal. One is to apply "finger test" on working board, dry finger first, push it over crystal area. A circuit with normal healthy margins should continue to function. The 32kHz oscillator is, however, a very delicate circuit, so even finger test might fail. (I've seen mass-produced designs where the crystal area is sealed by some compound).

The other test is to check the start-up time using a scope connected to Xout (not Xin !!!) pin of the circuit. For the 32kHz oscillator the start-up time should be between 1s and 5s. If you measure 5s, this is a sign of trouble.

To increase phase margins of the oscillator you can decrease values of loading caps, say, to 10pF, and see if the oscillator starts. If it will, it means that a different crystal needs to be selected, with smaller required load capacitance. Or, if +100ppm is not a problem for this design, you can leave it under-loaded.

For more guidance and application caveats, please consult with the manufacturer's appnote AN2867, or other nice explanations as this application report SLAA322C from TI.