Show that a decidable language is not decided by a decider in a given set M.Sipser's Introduction to the Theory of Computation offers the following problem in its chapter on decidability:

Let A be a Turing-recognizable language consisting of descriptions of Turing machines, {⟨M1⟩,⟨M2⟩, ...}, where every Mi is a decider. Prove that some decidable
language D is not decided by any decider Mi whose description appears in A.
(Hint: You may find it helpful to consider an enumerator for A.)

My qualm about this is that the question seems to imply finding a decidable language, the decider for which is not in the set of all deciders, which goes against the definition of decidability of languages.

Could you explain whether my doubts are justified? And if not, could you provide a proof (or a sketch of a proof) for the given problem (with or without the enumerator that is mentioned in the hint)?
Suppose that $A$ is a recognizable set containing descriptions of always halting Turing machines. We want to show that there exists a decidable language $L$ such that for any $M$ deciding $L$ we have $\langle M\rangle\notin A$.

We can construct such $L$ via simple diagonalization. Assume $A$ is infinite (otherwise the problem is trivial). We will define a Turing machine $M$ such that $L(M)$ does not have a decider in $A$. Let $M_A$ be an enumerator of $A$. Given input $x$, $M$ executes $M_A$ untill it outputs the $|x|'th$ word in the enumeration $\langle M_{|x|}\rangle$. Now, execute $M_{|x|}$ on $x$ and flip its answer.

I leave it to you to show that $M$ always halts and that $L(M)$ disagrees with $L(M_i)$ on at least $2^i$ inputs, where $M_i$ is the i'th Turing machine in the enumeration of $A$.