Why do constructivists not seem to care too much about call/cc
Constructive mathematics is not just a formal system but rather an understanding of what mathematics is about. Or to put it differently, not every kind of semantics is accepted by a constructive mathematician.

To a constructive mathematician call/cc looks like cheating. Consider how we witness $p \lor \lnot p$ using call/cc:


We provide a function $f$ which allegedly proves $\lnot p$. In reality $f$ is a bag of tricks.
If anyone ever applies $f$ to a proof of $p$, then $f$ unleashes call/cc to roll back time, and with a proof of $p$ in hand, changes its mind about $p \lor \lnot p$: this time claiming that it is a proof of $p$.


The constructive understanding of disjunction is algorithmic decidability, but the above is hardly making any decisions. As a test, a constructive mathematician might ask you how call/cc helps prove that every Turing machine halts or diverges. And what is the program witnessing this fact? (It ought to be the Halting Oracle.)