I think this post (from a math academy employee) has a good argument for why these sorts of exercises are important. It's about basic arithmetic, but I think it applies to tedious things like performing gaussian elimination on small matrices as well.

https://www.justinmath.com/if-you-want-to-learn-algebra-you-...

I like to come at it from both angles - higher level with useful applications, and then lower level "I could maybe implement this if I had to" exercises. The latter are tedious, and hard to motivate effort for without the former. Ultimately, as the post argues, I agree that if you don't understand the lower level (tedious) operations, you will only get so far in your ability to apply LA.