Can commercial pilot get paid for introductory flights with their own customers?
There are three parts to your question and the answer to all three is no. And there isn’t one specific place in the FARs that says you can’t do this, but reading Parts 91 and 135 along with appropriate ACs and Legal Opinions of the FAA leads most pilots to realize that they can’t do this.
§91.147   Passenger carrying flights for compensation or hire. contains a list of requirements that you must satisfy as well.

If you look for customers it is considered 'holding out' and according to AC 120-12A ‘A carrier becomes a common carrier when it "holds itself out" to the public, or to a segment of the public, as willing to furnish transportation within the limits of its facilities to any person who wants it.’ And if you are a common carrier, then for this type of operation, you must have a Part 135 certificate.

As a commercial pilot, you can be paid to fly someone in their airplane. Lots of commercial pilots do this. However, if you provide the plane then you are acting as a Part 135 operation and you can’t do it.

Getting paid is also not allowed. Since you can‘t do parts one and two of your question, you are acting under your Private Pilot certificate for the flight. At most, you can split the immediate expenses i.e. fuel, landing fees, etc. Even that is a bit dicey since you won’t have a ‘common purpose’ with the person you are flying. In other words, without holding out for customers, you wouldn’t have been making the flight anyway.

Not part of the question, but there are lots of things you can do with a commercial pilot’s license:

Corporate Pilot, Charter Pilot, Contract Pilot (for a private owner), Freight/Cargo Pilot, Ferry Pilot, Banner towing Pilot, Sight-Seeing/Tour Pilot, Agricultural Pilot/Crop Duster, Skydiver Pilot, Photographer Pilot, Airshow Pilot, Demo Pilot (aircraft sales), Medical Evacuation Pilot, Humanitarian/Charity Pilot, News/Traffic Pilot, Pipeline Patrol Pilot, National Defense or Border Patrol Pilot, Firefighting or Forestry Pilot.