The recognizability of $\overline{A}_{TM}$

In undergrad theory classes, the idea of decidability and recognizability is introduced. It's well known that $$A_{TM}$$, the set of words accepted by a TM $$M$$, is recognizable but not decidable. We also know that the complement problem $$\overline{A}_{TM}$$ is co-recognizable but not recognizable.

I have a doubt about the co-recognizability part. If $$\overline{A}_{TM}$$ is co-recognizable, then we have a TM that recognizes the set of words not accepted by some TM $$M$$. Doesn't that mean $$\overline{A}_{TM}$$ is recognizable, since we have a machine that recognizes the language?

I know this can't be the case, but I'm not able to find out where I'm going wrong with my logic.

If $$\overline{A_{TM}}$$ is co-recognizable, then we have a TM that recognizes the set of words not accepted by some TM $$M$$.
The fact that $$\overline{A_{TM}}$$ is co-recognizable means that there is a TM $$T$$ that recognizes all words not in $$\overline{A_{TM}}$$. Equivalently, $$T$$ recognizes all words in $$\overline{\overline{A_{TM}}}=A_{TM}$$, i.e., $$T$$ recognizes all words accepted by $$M$$.