First, no, you're not required to have a transponder to operate from controlled fields, so unfortunately, yes, you're quite mistaken. The only time you NEED a transponder for VFR flight is when you're within 30 miles of an airport that controls Class B airspace (like SeaTac, LAX, Frisco, etc). You can do negative-transponder flight into and around other airports, but the services available will be limited and some busier airports may say 'stay clear of our airspace' if things are really busy, but you're not required to have a transponder.
Second, controlled airports account for a miniscule percentage of airports in the US. Almost every airport is uncontrolled.
All of the PCAS systems use transponder responses to get traffic. If a plane doesn't have a transponder, then, well, it's not going to show anything. The PCAS may be battery operated, but it's worthless unless the other planes are using transponders.