As a practical matter, no. Legal told me I have to implement it for all users because it covers EU citizens who might be accessing our site from or living in non EU countries.

That doesn’t sound right. The EU doesn’t have jurisdiction in non-EU countries.

Schengen Agreement? I can go and visit a non EU country without getting rid of my rights.

European in terms of nationality is irrelevant, but I'm not sure if that's what you mean.

Most articles I've read on it state the user must be in Europe and taking a product/service delivered to Europe. A US tourist buying something in a store when they arrive in Europe has GDPR protection, a French tourist visiting the US (or even at home in France but buying something online to be delivered to a friend in the US) is not.

https://www.dyspatch.io/blog/gdpr-location-or-citizenship/

Right, so the equivalent here would have been if Blizzard had censored the player in China, but not in the rest of the world.

Correct. A judge just ruled that Google (I think?) did have to apply GDPR to countries outside of the EU