Has number theory affected the way you look at other branches of mathematics?

The quickest, least nuanced answer, is that your intuition is right: number theory, while very beautiful and a fruitful source of competitive exam questions, is not really a powerful tool in other parts of mathematics. In calculus, analysis, geometry, and topology, it's of quite limited utility. (I'm not saying *zero* utility, don't "at" me!) In abstract algebra it comes in a little handier ... Lagrange's theorem and Sylow theory spring to mind.

That said, since the early 1900s, there has been this thread of research relating number theory to analysis, and more recently, to algebraic geometry. A purely analytic conjecture, the Riemann hypothesis about the zeta function, has profound consequences for the distribution of the primes. And in the last few decades, crosstalk from the algebraic geometry theory of modular forms enabled Wiles to prove Fermat's "last theorem". But as far as I can tell (which might not be that far) most of the benefits flow one way: insights from elsewhere unexpectedly helping to knock over long-standing unsolved problems in number theory. Insights from number theory do not seem to be paying the rest of mathematics back, though if the bridge-building continues that may change.

None of this stops many really good mathematicians from studying number theory! It's beautiful stuff, well worth studying for its own sake. But I don't think you can make a really strong case for studying it on utilitarian grounds, as you seem to wish.

Number theory is neat for a lot of reasons. Depending on how far you go in your math journey, you’ll see that many topics in “applicable” math can be used to solve problems in number theory. But, to your point, number theory, aside from it’s application in cryptography and maybe some other things, is often seen as the crown jewel of math-for-math’s sake.
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Though number theory occasionally comes up in other areas, it's true that it doesn't come up particularly often.

However, elementary number theory is one of the few areas of math which are accessible for high school students to do proofs and solve problems in without knowing a lot of theory.

It is possible to learn to do proofs in algebra and calculus, but there are a couple of reasons that these tend to be less accessible. One reason is that the more interesting facts tend to be technically difficult to prove. Another is that some of the proofs involve facts that are already familiar and that the beginner won't necessarily see the point of proving.

As a result, elementary number theory is one of the main areas in which beginners are introduced to real mathematical thinking. The benefit in other areas of math is less likely to be in terms of specific facts you can use and more in having developed good habits regarding how you think about mathematical problems.
Elementary number theory is the basis for all abstract algebra, one of the most important fields of study in Math.

In Abstract Algebra we study sets equipped with some operations and properties. Most of those properties are generalizations of properties we study in elementary number theory. Besides that, most definitions of objects in abstract algebra are based on integers or rationals.

It is also important to say that Abstract Algebra has applications on other fields, as topology, for example.

To say that number theory doesn't come up often is not acccurate at all.
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