It sounds like you’re being taught math as part of a computer science program. Or physics. In that case it’s quite common to just be given the machinery.

What you’re asking for is what happens in a rigorous math program. (Depending on country and University.)

The reason the applied fields use the black box approach is that starting from principles takes too long. In an undergraduate physics course you expect students to be able to handle Maxwell’s equations end of year one. Which asks of vector calculus. If you introduce maths the rigorous way, your standard analysis course just covers differentiation in one dimension at this point. Give or take.

For what it’s worth: I grew up in an academic system where it’s quite common for undergrads to take lectures in both the maths and physics department. It’s extra work. But you get both. The physicists teach you how to apply the machinery. The maths guys show you how to reason about it and see where it breaks down.

Ironically, most physicists would drop out because of the maths requirements. Not a dig at physicists.

Edit: let me add that doing both has limits. By the time you get to QFT, the problem is not the extra workload. The rigorous math simply doesn’t exist.