If they had a mortgage on the property and the property was effectively signed over to the bank in the deed of trust, the bank would be the effective owner. The bank as a creature of state then would be subject to every law to which creatures of state would be subject. Furthermore, the 'borrowers' would likely be deemed to be servants/employees (ala debt) to the bank.
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The short-term rental "law" sound like the village is becoming as annoying as an invasive HOA.