In constructive mathematics I think you'd be right: to call a problem decidable would require you to produce the algorithm that decides it, so you couldn't call the Collatz conjecture decidable (nor could you call it undecidable!). But the usual definition of decidability is classical.
This is slightly bizarre now I think about it: the definition of decidability allows the algorithm selection to be undecidable!
This is slightly bizarre now I think about it: the definition of decidability allows the algorithm selection to be undecidable!