There are a lot of websites that only support third party login, so that is not always an option.
They don't have to bend for another, but they made a choice to put an app on iOS. They added support for apple signin, and then for some reason did not put it on their website.
You can criticize Apple for requiring that all you want, but they clearly have support for it and are choosing to not put it on their website which is causing a worse user experience.
IF apple did not support website loggin than sure, but they do. So the ability to fix this is on Anthropic (and many other websites).
If you are already going to support third party login you should not limit it to only Google accounts and there is no reason to support Apple on iOS and not the web.
Also for the record, Apple only requires sign in with apple if you already support third party authentication. So if you are already going to support that, giving the user more choice (and making it so we are all a bit less dependent on google) is a good thing.
No criticism from me towards apple or Anthropic. Both parties made their choice. Apple was late to the identity business and the other ships had already sailed.
Third party logins are an extension and a massive risk to any website that doesn't include email hosting.
We have see identity providers dissapear, and people may change their mind.
Easiest way is to register you rown domain and use it with an identity provider of your choice and be able to move it anywhere.
Otherwise we are a faceless citizen of a corporation that can handle access to our identity and everything attached to it without recourse or access to anyone.