Yes, I think typical intro classes focus too much on the rows and not enough on the columns. We learn about change of basis where each element in the new vector

Yes, I think typical intro classes focus too much on the rows and not enough on the columns. We learn about change of basis where each element in the new vector is a dot product with a row of the matrix. But you can also look at the new vector as a weighted combination of all the columns in the matrix. You mentioned opengl and it's a perfect example of why you need to learn both. The Model matrix is column-oriented - if Z is up in the model but Y is up in the world, you need a Y unit vector in the third column of the matrix. The View matrix is row-oriented - each row is an axis of the camera's orientation. The columns perspective also makes it easier to understand homogeneous coordinates.