There are important pieces of mathematics. Anyone claiming that differential equations are a niche interest is lying to you. And then there are niche interests. These are worthwhile fields. It’s just you can get a good well-rounded mathematical education while being only a little aware of them. And things can move from being important to niche, or back again.
Continued fractions are one of those things I had understood to have fallen from importance. They had a vogue, in Western mathematics, where they do some problems pretty neatly and cleverly. But they’re discussed more rarely these days. The speculation I’ve seen is that they don’t quite have a logical place, as being a little too hard when you’re learning fractions but seeming too easy when you’re learning infinite series, that sort of thing. My experience, it turns out, was not universal, and that’s an exciting thing to learn in the comments.