Both linear algebra and calculus are classic introductory math classes. A lot of calculus IS linear algebra, given that derivations are linear approximations.
I had three years of high school with a substantial amount of calculus; by contrast while I was introduced to matrices and Gaussian elimination at high school, the treatment of the algebraic side was almost completely superficial and focussed on solving systems of linear equations.
At university that changed, with a clear treatment of vector spaces and linear algebra as generally interesting topics. But still, they received far less time than calculus in my first year in a maths degree.
Now that was over 30 years ago and things might be a bit different today. But I have the impression that the more generally applicable subject of linear algebra gets less time unless students get interested in the applications that demand it be taught properly.
The equations you quoted result from minimizing the square of the norm of the residual of Ax-b over all inputs x, so in a sense least squares is just calculus…
Transposes correspond to integration by parts, and the question of whether A^T*A has an inverse can get involved. Also for infinite matrices the analogies hold more readily, see the observation due to Alan Edelman on page 8 of https://klein.mit.edu/~gs/papers/Paper5_ver7.pdf
I’d agree, but most first year students aren’t ready to work at the level of abstraction required for a rigorous LA course (i.e., more than just matrices and Gaussian Elimination etc.)