1: Up until the collegiate level, most math students memorize the 'right' way to do certain types of problems, and then repeat that hundreds of times on homework, so that they can repeat it on a test.
However, with integration there are too many possible forms to recognize and memorize each one. Instead the student is forced to try to figure out their own way of solving the problem from a common set of rules. Even though this process can often be automated, there are similar reasoning tasks, such as proof-writing, which are impractical to automate at this time. There are reasoning tasks in other fields such as programming and writing which also require the same kind of thinking: manipulation by predefined rules.
This does not always have its intended effect, as many students are stuck in the memorization-mindset, and it is easier for professors to say what to do without explaining why or what it means. There is (unfortunately) still much memorization involved, but it asks for more original thinking than prior math questions have.
2: For approximations, people try not to approximate things that could be computed directly to get better accuracy; approximations are usually for things that can't be integrated analytically.
Knowing what an integral really means (as opposed to knowing how to operate a CAS) is both useful and important even if students do not become mathematicians. The concepts of integrals and other infinitesimals lay the foundation for differential equations and stats. These are heavily utilized by all STEM fields, business, and the social sciences.
Rather, they should be viewed as two complementary skills that are valuable in different ways: learning how to use a CAS, and learning the theory (what it really means, why it is that particular way).
3: Maybe "integrate this function" problems gets boring and distant. It feels really useless if you could just type it into a CAS. Good professors will ask questions whose difficulty does not come from bookkeeping symbols but from figuring out what to do next.
Example: explain why the area of a circle is $\pi r^2$.
Try to answer this question on your own.
If you imagine the circle like an onion slice with concentric rings of width $dr$ (just a weird looking name for a small number), then you could 'unroll' each ring into a rectangle whose length would be the circumference of that ring $2 \pi r$ and width would be $dr$. Adding up all of the rings, and voilà $\int 2 \pi r\ dr = \pi r^2$.
Or maybe don't teach integral calculus at all
I have answered thus far as a 'Devil's advocate', and I am sympathetic to your viewpoint. I think there is some use to learning higher-level math so that students become more logical thinkers, but maybe not integral calculus. Integration involves more memorization than I would like, and ends up being tedious rather than rewarding.
I would be happy to see a number theory course be offered in place of calculus. Number theory asks questions like "how many prime numbers are there?” These questions sometimes have counterintuitive and interesting answers.