The “SAT reasoning math” section tests reasoning in math. This is given to high school students.
Stanford uses this test to assess the ability to reason in math.
Thus math thinking does occur prior to college is the official position of the SAT and of Stanford.
Math reasoning does not mean having no difficulty in problems or understanding. Resolving difficulty is very much part of mathematical thinking. Mathematical thinking is spent a good deal of the time having difficulty understanding something. This can stretch over years. Poorly written or conceived materials, especially from the math establishment, may be part of that problem.