> It doesn't help me to spend a week on defining and proving integrals without first understanding that the goal is to find the area of a rectangle or circle
I will only discuss this point (but for your other points similar arguments can be made). So you want to measure some n-dimensional volume? This is a highly non-trivial problem, since it is quite possible to find subsets of R^n where you can't assign volumes in a sensible way:
> https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
If you don't care about this subtlety you end up in such a paradox.
So you think hard about how you can define "measureable" in an accurate sense and come up with the definition of a sigma algebra and think about how you can even define measures (say of volume) on a sigma algebra. Because these can be ugly to describe, you will immediately think about how one can describe the sigma algebra and the measure in a more easy way. So you come up with generators and note that the measure is defined uniquely by its value on the generator.
Doing this process on the real line using the Borel sigma algebra/measure is not that complicated (so you solved the problem for the real case). From here it is only a small step to the Lebesgue-Stieltjes integral in R^1 that is really worth doing.
But now you remember that you wanted to compute higher-dimensional volumes. So you ask how can you recycle what you've already done? The answer is obvious: You define product sigma algebras and product measures.
As you have seen (Banach-Tarski_paradox) all this is necessary just to even define volumes. I really know no easier way that is both useful in practise (at least be able to obtain the volume of spheres (i.e. not just "simple objects as simplices")) and avoids the problem that was laid out by Banach and Tarski-