What do you mean by "often better". Has it even ever been tried out?
Of course, and it's standard practice in some places.
P.S. I'm talking about extracurricular math clubs/circles. School curriculum is a separate thing that's divorced from actual math.
By definition, quantities, either discrete or continuous, as already classified by Aristotle, are the properties that can be expressed by numeric values, as a result of counting for discrete quantities, or as a result of measuring for continuous quantities.
Properties that are not quantities, but which can be compared, to determine which is greater or less, or if they are equal, are qualities.
The simplest numbers, i.e. cardinal numbers a.k.a. natural numbers, are equivalence classes of sets, so indeed the theory of numbers is built upon that of sets (a.k.a. multitudes in older language), but this does not contradict in any way the title from the book, because quantitative thinking refers to only a part of mathematics, not to all of ite.
This isn't true though. It's not supported by anything except Bronze Age tradition.
I.e., when teaching kids advanced math at an early age it's often better to skip numbers altogether and start with set theory or logic. Many kids find that a much easier gateway.