There is an interesting class of problems that are provable in $ZF$ by "coincidence" in the sense that either (a fragment of) $AC$ holds and then it is provable from the proof in $ZFC$, or there specific instance of the failure of $AC$ implies the result.

The simplest example for such problem is: countable product of spaces with $2$ elements is compact.

If the product of our spaces is non-empty, we have fragment big enough of $AC$ to prove that the product is compact by following the proof in $ZFC$, otherwise the product is empty, but the empty space is compact, so we are done.

A more complicated example is: let $X$ be subspace of $ℝ$, then $X$ is compact if and only if $X$ is sequentially compact and Lindelöf.

If $CC(ℝ)$ holds then the usual proof works, on the other hand $¬CC(ℝ)$ is equivalent to "subspace of $ℝ$ is Lindelöf if and only if it is compact", so because $X⊆ℝ$ is Lindelöf, it is compact.

The proof that $¬CC(ℝ)$ is equivalent to "Lindelöf ⇔ Compact" for subspaces of $ℝ$ is, as far as I know, not a trivial result.

There is an interesting class of problems that are provable in $ZF$ by "coincidence" in the sense that either (a fragment of) $AC$ holds and then it is provable from the proof in $ZFC$, or there specific instance of the failure of $AC$ implies the result.

The simplest example for such problem is: countable product of spaces with $2$ elements is compact.

If the product of our spaces is non-empty, we have fragment big enough of $AC$ to prove that the product is compact by following the proof in $ZFC$, otherwise the product is empty, but the empty space is compact, so we are done.

A more complicated example is: let $X$ be subspace of $ℝ$, then $X$ is compact if and only if $X$ is sequentially compact and Lindelöf.

If $CC(ℝ)$ holds then the usual proof works, on the other hand $¬CC(ℝ)$ is equivalent to "subspace of $ℝ$ is Lindelöf if and only if it is compact", so because $X⊆ℝ$ is Lindelöf, it is compact.

The result of $¬CC(ℝ)$ being equivalent to "Lindelöf ⇔ Compact" for subspaces of $ℝ$ don't, as far as I know, have a simple proof.

Herrlich, Horst, Products of Lindelöf (T_2)-spaces are Lindelöf – in some models of ZF., Commentat. Math. Univ. Carol. 43, No. 2, 319-333 (2002). ZBL1072.03029.