USA - All students must show "papers"

Where in this does it state that an existing pilot certificate is all
that is required???

Any U.S. citizen applying for training, including recurrent training,
in an aircraft weighing less than 12,500 pounds must present the
flight school or flight instructor with evidence of U.S. citizenship.
Evidence may be shown by one of the following:
Valid unexpired U.S. passport
Original birth certificate and government-issued picture ID
Original U.S. naturalization or citizenship certificate with raised
seal and government-issued picture ID
Original certificate of U.S. citizenship and government-issued picture
ID
The flight school or flight instructor will retain a copy of the
documentation for a period of 5 years.
Impact to flight schools and freelance flight instructors:

Any flight school or a freelance flight instructor providing training
in an aircraft weighing less than 12,500 pounds must comply with the
following:
Determine whether an individual is an U.S. citizen. Evidence may be
shown by one of the following:
Valid unexpired U.S. passport
Original birth certificate and government-issued picture ID
Original U.S. naturalization or citizenship certificate with raised
seal and government-issued picture ID
Original certificate of U.S. citizenship and government-issued picture
ID
Register with TSA if providing training to foreign students
Notify TSA when a foreign applicant requests training
Submit to TSA a photo of the foreign applicant after he or she first
arrives for training
For recurrent training, the following must also be submitted to TSA:
Foreign applicant's full name
TSA/DOJ identification number
Copy of foreign applicant's current, unexpired passport and visa
Training details
Photo of the foreign applicant after he or she first arrives for
training
Retain applicant information and TSA approval records for five years.
The records are subject to TSA audit.
Immediately terminate a foreign applicant's training if informed by
TSA the applicant poses a threat to aviation or national security.
Ground school and demonstration flights are exempt from the rule and
recurrent training is exempt from the fingerprints requirement.
Flight schools and freelance flight instructors, regardless of whether
they are training foreign students, must provide initial and annual
recurrent "security awareness training" for each flight school
employee that has a direct contact with a flight school student
(regardless of citizenship or nationality).
Flight school employees must receive the initial security awareness
training by January 18, 2005. Employees hired after January 18, 2005
must receive the training within 60 days of being hired.
Schools must maintain a record of such training for one year after the
employee leaves the school. Again, these records are subject to TSA
and FAA audit.
TSA's initial online security awareness training program will be
available on October 30, 2004 at www.tsa.gov.
Flight schools, including freelance flight instructors, that fail to
comply with the rule's requirements may be subject to enforcement
action.