Sure appraiser checked the wrong box but I don't see how that matters. Appraiser's are only required to call for further inspection ONLY if there were readily observable issues discovered during the walk through. At least you, the agent, the underwriter or all three knew there was a septic and a well on this property. Having a well and a septic on same property should have been a redflag for the lender which would have prompted a septic inspection and a survey. Both of these should have been provided to the appraiser before he even came out to the property. Marking it public or private is inconsequential.
By the way, the book you are quoting from is at least a decade old and is not the same handbook that appraiser's refer to. Please see latest copy of Handbook 4001.1 dated 10/26/2021.
Here are a few excerpts from the most recent handbook:
Mortgagees bear primary responsibility for determining eligibility and the sufficiency of collateral; however, the Appraiser provides preliminary verification that the Property Acceptability Criteria have been met and an appraised value for the property.
The Appraiser must obtain all of the following from the Mortgagee before beginning an appraisal: surveys, ... any other legal documents (IE, home inspection, etc.).
The Appraiser must identify readily observable defective conditions.
Appraisers may not recommend inspections only as a means of limiting liability. The reason or indication of a particular problem must be given when requiring an inspection. (IE, if an appraiser does not have evidence there is a problem with the septic, he/she is not required to call for further inspection, lender may opt to do so).
If inspection is required, the Appraiser must cite the reason for requiring an inspection.
The Appraiser must flush the toilets and operate a sample of faucets to observe water pressure and flow, to determine that the plumbing system is intact, that it does not emit foul odors, that faucets function appropriately, that both cold and hot water run, and that there are no readily observable evidence of leaks or structural damage under fixtures.
If the Property has a septic system, the Appraiser must visually observe it for any signs of failure or surface evidence of malfunction. If there are readily observable deficiencies, the Appraiser must require repair or further inspection.
The Appraiser must also be familiar with the minimum distance requirements between private wells and sources of pollution and, if discernible, comment on them. The Appraiser is not required to sketch or note distances between the well, property lines, septic tanks, drain fields, or building structures but may provide estimated distances where they are comfortable doing so. When available, the Appraiser should obtain from the homeowner or Mortgagee a copy of a survey or other documents attesting to the separation distances between the well and septic system or other sources of pollution.
Hope this helps in determining who is to blame for not getting a septic system inspection. Good luck.