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I think Strang has some good points, and his advice is true for the majority of students who won't go on to use substantial amounts of mathematics or need to learn much math in the future. However, if you do want to learn more math beyond calculus and linear algebra, then a thorough understanding of calculus is absolutely a prerequisite and you honestly probably haven't learned it well enough taking the typical sequence of classes.

For example, I'm currently studying probability theory at a fairly rigorous level (building up from measure theory) and my relative lack of skill with calculus is the main stumbling block. And I was recently talking with one of my professors who said that the biggest obstacle he sees students facing in trying to understand advanced mathematics isn't the difficulty of the math itself, but in their lack of mastery of the fundamental calculations that are necessary to follow the ideas and to come up with them on your own. Fluency with calculus is an enormous part of this. That said, I agree with the point, which is that for most students linear algebra is more useful.

Actually, I might go a bit farther and say that most students don't need to learn much math at all, at least in the way it is taught. Math is pretty poorly taught for a subject that is actually quite interesting once you start to understand it better (part of the problem is that in order to get to the interesting parts you need to first master all the boring stuff), and most people won't use any nontrivial math in their future lives.

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This may be true for mathematics in the analytical tradition (things like Topology, Measure Theory, Real/Complex Analysis), but Linear Algebra is far more important in, unsurprisingly, algebraic disciplines. These include Number Theory, Field Theory, and Mathematical Logic.

While studying mathematics in college, once I had finished my Real Analysis requirement, I jumped headfirst into the algebraic side of things and never found myself using any sort of calculus. Even in my Topology course, we focused much more on using techniques from Real Analysis than specifically calculus topics (the former just being a generalization of the latter).

Probability theory is somewhat deceptive in its classification, since much of it "feels" a lot like a discrete mathematics course; however, much of the concepts, like you say, are underpinned by measure-theoretic principles, which is heavily analytic. It makes sense that calculus would come in handy in a much deeper study of probability theory.

That's a good point, my own studies have been biased towards the analytical side of mathematics, but I can definitely see how calculus could be less useful in other fields, e.g. I don't remember ever needing to do an integral when learning algorithms. And the opinions of my professors are of course biased as well, since they've spent their entire careers on analysis.

When analyzing algorithms, I have used calculus as a shortcut to many calculations. I believe that learning calculus is not just about its direct applications. Calculus often comes up when just dealing with your set-up.

You might make a discrete model for a problem and end up using calculus for approximations, equivalent calculations, or reasoning.