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The logo is probably just supposed to another opportunity for them to grab mind share by hosting the site for free and sticking that there, but it creates a confusing impression.
Out of curiosity, how is Better Auth funded?
Clerk sponsors it extremely heavily so that when you get fed up with dealing with it, every single page has a CTA funneling you to them.
Honestly the whole situation has a terrible terrible smell, but that's post-Rauch JS for you.