So, there are two answers to your question.

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.