Note the difference between referring (or recommending) your friend, and getting him the job. The latter implies that you influenced the company to make sure they would hire him; which would be perceived badly.
However, if you merely referred/recommended him to the company for a known vacancy, and the company independently interviewed and hired your friend; then you have done nothing wrong.
Some people might think that it to be unfair because he didn't really find the job on his own merit.
It's not about finding the job. The skill is not in managing to find the vacancy. The skill is in getting the company to agree that you are the right employee for the vacancy and offer to hire you. This is not impacted by how you came to know about the vacancy in the first place.
All you did was point out that there's a vacancy. You did not in any way make the interview easier for them than it would've been for others (I assume).
On the other hand, not admitting it might look like if I were hiding something relatively meaningful
Yep, it suggests a guilty conscience. Even if there is none, it will be inferred.
Maybe I am overthinking this?
I think you are.
Simply make sure to not imply that you got him the job. That implies that you did unfairly benefit your friend in the hiring process.