I think it's manifest that they aren't implying that. They said directly that the pun is the part they found objectionable.

I think it's manifest that they are. This person doesn't find puns as a linguistic tool to be offensive. Their angle is that the use of a Korean name to address a Korean person with that name makes this pun racist.

If I made a pun about a guy named John is that racist? The only way that could be justified is if you somehow think it's shameful to be named John for some weird racial reason.

I’m not interested in a factual dispute over what the GP claimed. I’m only pointing out that they clearly took issue with the pun, and not the name itself.

The pun is the name. What else in the pun is there to be offended by except the name itself? I would seriously like to understand this.