I completely agree with you. A corollary of my comment is that while not necessarily providing theoretical justification, sometimes doing this whole problem solving thing helps seeing why a field of knowledge came to study some of its problems, fast. For example, I have a strong abstract foundation in Functional Analysis, Measure Theory and Optimization and my field is Numerical Analysis of PDE. I didn't understand 1. some mechanical properties of materials, 2. some properties of tensors and differential geometry that I needed. But I didn't have so much time to devote and work out the proofs for all the abstract apparatus on my own. So I took an engineering textbook on tensors, some notes on relativity (because they use tensors and Christoffel symbols) and in a few weeks I had almost understood everything I needed.

On the other hand, there is an extremely rigorous book with all the proofs for differential geometry, by Kennington. It has 2400 pages. For tensors I would guess that I need more Algebra, specifically, modules. And to review smoothness and differentiability classes.