> but I don't think that affects the definition of a public good

You happen to be misconstruing the two definitions of "public good":

> a commodity or service that is provided without profit to all members of a society, either by the government or a private individual or organization.

> the benefit or well-being of the public.

As such, something can be a "public good" without being "a benefit".

[0] https://www.google.com/search?q=public+goods&si=AMnBZoHHbOut...