To people whose experience of mathematics does not extend far, if at all, beyond the high school math class, I think it’s actually close to impossible for them to really grasp what mathematical thinking is.
If high school math covers algebra, geometry, analytic geometry, functions and calculus, then it covers parts of math from antiquity up to some in the 19th century.
If we consider pre-19th century algebra, analytic geometry and calculus as procedural and not conceptual, then we would exclude Euler, Newton, Fermat, Lagrange, etc. from thinking mathematically.
Certainly Euclid is excluded.High school math geometry texts are considered more rigorous than Euclid.
Even if students find the epsilon delta definition difficulty, they do read it, and Newton did not.
Set theory including ordered pairs, functions and relations are covered in a way that is 19th century even if only some of it is covered. This was certainly not known to the 17th century.
So Euler, Newton, Fermat and Largrange did not only not do mathematical thinking, they could not even grasp what it is.
And a high school physics course that covers a little relativity? That isn’t physics thinking either.