You are correct, the regs don't state that you need training in a light-sport aircraft. It only states that the applicant should receive 20 hours of flight time, including 15 hours dual from an authorized instructor in a single engine airplane and at least 5 hours of solo flight.
... BUT the holder of a sport pilot certificate may only act as PIC of a light-sport aircraft. A sport pilot may operate any make and model of LSA in the same category and class and within the same set of aircraft as the make/model in which he/she is endorsed.
As a sport pilot student, the student also may not act as PIC on any aircraft other than a light-sport aircraft, or at night, at an altitude more than 10,000 ft MSL or 2000 ft AGL (whichever is higher) or in Class B, C, or D airspace or at an airport with a control tower with receiving the required ground/flight training and endorsement from an authorized instructor.
We've had a couple people come to our flight school inquiring about a sport pilot certificate, basically we tell them that we don't have an LSA aircraft but will be happy to work with them towards a recreational pilot certificate. A recreational pilot certificate is cheaper than a private and the recreational pilot with a category and class rating can operate any light-sport in that category and class. If, however, the person wants to get sport pilot certificate that bad, refer them to a flight school that has an LSA aircraft and provides sport pilot training.