you need one of those new fancy GitHub EMU enterprises instead of a normal one. it is driven solely by an identity provider like Okta or AzureAD.
no one outside of your identity provider will ever know about you. your users have a read-only view of the greater github.com, but whatever access you give them within your enterprise.
it's impossible to make anything public in a GitHub EMU enterprise, and your users won't even be able to star repos outside of the enterprise, because that would reveal your presence.
if you can live without any public access to your stuff at all, have a look. you can convert your old enterprise org(s) to a Teams subscription and continue to publish open stuff there, but you'll need personal accounts there, like always.
no one outside of your identity provider will ever know about you. your users have a read-only view of the greater github.com, but whatever access you give them within your enterprise.
it's impossible to make anything public in a GitHub EMU enterprise, and your users won't even be able to star repos outside of the enterprise, because that would reveal your presence.
if you can live without any public access to your stuff at all, have a look. you can convert your old enterprise org(s) to a Teams subscription and continue to publish open stuff there, but you'll need personal accounts there, like always.