It is possible that the wood had been covered with lead paint and the paint was stripped.

But if you test things that are not listed on the kit - who knows what you will find. It may be reacting with the tannin in the wood.

Definitely could have been the case. From safety perspective, I wonder if I should be replacing the wood, or just strip again, prime, and paint.

I doubt the stripped wood would create a lead "problem". Lead is only an issue if ingested or airbourne, neither will happen now that the paint has been removed. I'm not an expert but I'd research this much farther.