This statement is correct only on a very basic, fundamental sense, but it disregards the research practice. Let's say you're a mathematician who studies analysis or algebra. Sure, technically there is no fundamental reason for constructive logic and classical logic to "compete", you can simply choose whichever one is useful for the problem you're solving, in fact {constructive + lem + choice axioms} will be equivalent to classical math, so why not just study constructive math since it's higher level of abstraction and you can always add those axioms "later" when you have a particular application.
In reality, on a human level, it doesn't work like that because, when you have disagreements on the very foundations of your field, although both camps can agree that their results do follow, the fact that their results (and thus terminology) are incompatible makes it too difficult to research both at the same time. This basically means, practically speaking, you need to be familiar with both, but definitely specialize in one. Which creates hubs of different sorts of math/stats/cs departments etc.
If you're, for example, working on constructive analysis, you'll have to spend tremendous amount of energy on understanding contemporary techniques like localization etc just to work around a basic logical axiom, which is likely irrelevant to a lot of applications. Really, this is like trying to understand the mathematical properties of binary arithmetic (Z/2Z) but day-to-day studying group theory in general. Well, sure Z/2Z is a group, but really you're simply interested in a single, tiny, finite abelian group, but now you need to do a whole bunch of work on non-abelian groups, infinite groups, non-cyclic groups etc just to ignore all those facts.