I would say it really depends on which cap was reversed and the role it plays in the circuit. A reversed cap in a power supply with high voltages usually pop/explode right away. So that would be an extreme example that fails right away.
But I can speak from personal experience when I bought a just-recapped Mac preamp. For the seller, it operated fine, but I don’t think it was really used after recap, just briefly checked. After I first received it, it played fine at first too.
But within the first couple/few days, it started slowly muting itself after being on for a bit. Took a lot of troubleshooting, but finally traced it to a reversed cap in the muting circuit.
I replaced the cap with a new one in the correct polarity/orientation and never had the problem again.
In the case of your power on self-check circuit, it’s verifying that voltages in certain parts of the circuit are in the expected/proper range. If a cap is in the wrong way, it could be “fighting” what the circuit is trying to do, dragging a voltage up/down from where it’s supposed to be. Whether the wrong polarity cap slowly degrades from being reverse-biased and eventually has different characteristics or fails and no longer has the same effect as it did when first implemented- that seems possible to me.
So yes, I think it’s possible you’re still dealing with a reversed polarity cap somewhere in the circuit. Not to say it’s 100% the case, but it’s worth checking to avoid chasing your tail over a possible issue that can easily be solved.