Why is a modal analysis of a very small MEMS piezoelectric resonator missing
modes and taking too long?

Your system of uMKS may be giving the solver problems due to
small mass density values in Kilograms/cubic micron. I would like to
propose a different unit system:
uM (1e-6 M) pKg(1e-12 Kg) us (1e-6 s) uN (1e-6 N).
With this new system the solver quickly finds 100 modes and the
frequencies match the values you were getting (except they are now in MHz.)
Use of the MKS system also works.

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