This is not a question without questions, but I must say that I think your logic is suspicious. You should not think about alternative authentication solutions, but the recently announced ASP.NET vulnerability should not force you to abandon the current (supposedly working) solution. I'm also not quite sure what the relevance of this comment is:
From what I understand, Microsoft typically stores things on the client side, because it simplifies working with server farms without requiring database access calls.
What is this vulnerability that makes you think that ASP.NET auth forms are broken more than another solution?
The details of the MS consultant seem to suggest that virtually any other authentication system could be exposed to a similar attack vulnerability. For example, any solution that uses the web.config to store settings will still have its own settings for the world, suggesting a successful attack.
The real solution here is not to change the security, but to apply the published workaround to the problem. You can switch authentication providers only to find that you are still vulnerable and your efforts won nothing.
Regarding tokens / sessions: you need to click something on the client for authentication (whether you call it a token or not), and this is not this part of the process that causes the current security problem: this is the way the server responds to certain calls that make this The secret is vulnerable to attack.
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