This is a very good, astute question. To paraphrase, and pointedly so, the question can be "what do we want an integral to do?" Yes, for example, as in the question, what should an extended notion of integral be compatible with? Certainly all elementary things, e.g., fundamental theorem of calculus, areas under curves, and so on. But/and how can we reasonably capture all this in a rigorous way?
One way, and not the only, and not the only thing we'd want, is a sort of continuity on continuous (scalar-valued, for example) functions of compact support. An easy version of the Riemann integral shows how to integrate such functions very well, compatibly with the fundamental theorem of calculus (and, by design, with area-under-curve computations).
It is not entirely trivial to give the space of continuous, compactly-supported (scalar-valued) functions on a reasonable space (e.g., $\mathbb R$) the correct topology... but if we do so, then the Riesz-Markov-Kakutani theorem says that any continuous linear functional on that space is given by "an integral (against a positive, regular, Borel measure". Good!
And ... uniquely so...
It is certainly true, though, that this characterization does not directly address issues about differentiation in a parameter under the integral, and so on. I would claim that such issues are best addressed thinking in terms of Gelfand's and Pettis' ideas from 1930s, about "weak integrals". Not elementary, but really on-target.
If you have follow-up questions, please do. I've thought about such issues for quite a while now, as they do play a significant role in much of mathematics, even while being dubiously dismissed as "folkloric" or "it's just the definition".