You're going to have a hard time convincing me that... vectors are a necessary precursor for regression analysis...

So you're fitting a straight line. Parameter estimates don't require linear algebra (that is, vectors and matrices). Super. But the immediate next step in any worthwhile analysis of data is calculating a confidence set (or credible set, if you're a Bayesian) for the parameter estimates; good luck teaching that if your students don't know basic linear algebra. In fact, all of regression analysis, from the most basic least squares estimator through multilevel/hierarchical regression models up to the most advanced sparse "p >> n" method, is built on top of linear algebra.

(Why do I have such strong opinions on the subject? I'm a Bayesian statistician by trade; this is how I make my living.)