Yes, I think it's good to learn math and read math textbooks in a "big picture first", coarse-to-fine manner. Before you learn your way around a city, you first look at a map of the earth to decide which city you want to visit.
I think usually reading the entire textbook thoroughly may not even be the right goal (unless the book is fundamental to your research area and you really need a deep mastery of it). The ocean of knowledge is infinite. You can never understand all the drops of water in the ocean, but you can soar over the water like a seagull, occasionally diving down to catch some prey.
Here's a description of how the mathematician Peter Scholze (who is said to be revolutionizing arithmetic geometry) learns math:
At 16, Scholze learned that a decade earlier Andrew Wiles had proved
the famous 17th-century problem known as Fermat's last theorem, which
says that the equation $x^n + y^n = z^n$ has no nonzero whole-number
solutions if $n$ is greater than two. Scholze was eager to study the
proof, but quickly discovered that despite the problem’s simplicity,
its solution uses some of the most cutting-edge mathematics around. “I
understood nothing, but it was really fascinating,” he said.
So
Scholze worked backward, figuring out what he needed to learn to make
sense of the proof. “To this day, that’s to a large extent how I
learn,” he said. “I never really learned the basic things like linear
algebra, actually — I only assimilated it through learning some other
stuff.”
Elon Musk, who has created a grade school called Ad Astra, makes some interesting related comments in this video.
Let's say you're trying to teach people about how engines work. A
more traditional approach would be to say, we're going to teach all
about screw drivers, and wrenches, and you're going to have a course
on screw drivers, a course on wrenches, and all these things, and that
is a very difficult way to do it. A much better way would be like,
here's the engine, now let's take it apart, how are we going to take
it apart? Ah, we need a screw driver, that's what the screw driver's
for. We need a wrench, that's what the wrench is for. And then a
very important thing happens, which is that the relevance of the tools
becomes apparent.
Richard Feynman mentioned that he quickly skims the whole book to get the big picture and see how the ideas fit together, before digging in to the detailed arguments. (I can't remember where Feynman said this.)
