They are the department that records deeds, liens, mortgages, and a lot of other stuff into the "public records".

I am enrolled in Public Service Loan Forgiveness (PSLF) program and need to pay 10% of my taxable income towards my student loans.

If for any reason, the Prospective Tenant(s) fail to complete all move-in requirements by the above listed date, the Landlord will return the property to public offering and the entire binder sum will be forfeited to the Landlord for expenses including, but not limited to, lost rent, holding costs, advertising costs, and marketing costs.

So we had to prove the unit was existing for more than 20 years but providing public records and voter logs and getting neighbors to sign statements.  

Is there a website that shows this kind of information publicly?

Most will likely stick to the technical limit of Sept 2021 when it ceased being a primary residence, but some may be comfortable going by the address on the 2021 return and others may explore a partial exclusion, depending on the reasons for your move abroad.If you'd like to read up on the technical rules, the IRS publication 523 gets into more detail than most people would like to know, but it does have some helpful tables and examples for different scenarios.

I am unsure what case law there may be regarding STRs with regards to "public accommodations" versus a typical rental arrangement.

Probate is public, requires paperwork, and changing of the Title/Deeds. 

The following is in my purchase agreement and I alway get a MOC on all purchases  MEMORANDUM OF CONTRACT RECORDABLE, PERSONS BOUND AND NOTICE: Buyer may cause to be recorded, at Buyer's option and expense, in the public records of the county in which the property is located, an executed Memorandum of Contract.

Would absolutely verify employment and do credit and background checks, including public records.