1542 applies to all airports requiring a security program or any airport that serves an operator that must have a security program. So, in short, yes, all airports are subject to TSA supervision (there are probably a very few small, privately-owned/personal airfields, and military fields, that are not covered).

But your larger point is well taken and some correction/clarification is needed, particularly in regard to TSA as an agency vs TSOs as employees.

TSA as an agency has extremely broad authority to supervise/oversee and inspect/ensure compliance with security requirements at airports.

Generally this authority is applied to only the sterile, SIDA and AOA areas of an airport. However, this authority does extend to areas adjacent to the sterile, SIDA and AOA areas. So, TSA has authority over extensive (but not "all") areas of airports including parking areas that may be next to, or just near, a SIDA or AOA.

TSOs (as differentiated from TSA as an agency) have little or no authority to search beyond the area of their post which could be at passenger, baggage, or cargo screening points, or entry points to certain areas of the airport, unless otherwise established by agreement between the airport/aircraft operator and TSA. And as has been discussed many times, when a TSO discovers a non-compliant situation it is the law enforcement agency having jurisdiction that resolves the situation.

But TSA employees from various offices (Inspections, Security Operations, Law Enforcement) have broad authority to carry out the broadly-defined statutory responsibilities of the agency and this includes inspections/searches.

Having said all of that - you are correct that those TSA employees who are authorized to inspect cannot arbitrarily stop anyone and conduct a search anywhere on the airport property; this limitation also applies to the local law enforcement agency. As you said, outside of designated areas they must have a reason to stop and search someone.