Visa's are entirely separate from the TSA clearance. For Visa issue your school must be approved by immigration, and they only approve 141 schools.
The students who can be taught under part 61 and require TSA approval are those who have legal status from another avenue, (such as permenent residents with a green card) rather than those seeking a visa for flight training. It would also include those seeking training for FAA certificates abroad.
You can give FAA training on a tourist visa, or a visa waiver provided that the training is less than 18 hours of study per week, and that it is incedental to the visit. For example, someone who is already a pilot visiting to build hours, who also intends to get a multi-engine add-on while he's here, may not require a visa, but would require TSA approval. If the course of study is less than 18 hours per week, then they are not eligible for a student visa.