To each their own, but I think there is value in seeing them multiple ways. The rows can be looked at as a system of equations. The columns can be vectors to be combined. The matrix as a whole can represent the covariance of a Kalman filter or describing a confidence ellipsoid. They can be an adjacency matrix describing the edges of a graph. They can be your entire data set, and so on. Some of these uses feel more like operands than operators.

Is there an online course that covers linear algebra in that way?

Gil Strang taught 18.06, the more calculation-focused and elementary version of linear algebra, when I was at MIT. The approach mentioned by the OP was what was taken in the more theoretical version, 18.701. I think that class has notes and lectures on OCW.

Not an online course, but Linear Algebra by Serge Lang is nice little book that is in this spirit (good to look at if you’re familiar with LA basics).

3blue1brown has great linear algebra videos that take it from the transformations point of view.