Is it possible for an academic to successfully claim their Google knowledge panel with an institution-hosted webpage and a Google Scholar account?
After multiple emails with Google support who didn't seem to understand the problem (or didn't want to), I finally obtained a clear answer: currently their verification process doesn't even consider that showing ownership of the Google Scholar account is a valid proof:


Sorry but you didn't answer the question: is Google Scholar a valid profile for the verification process? yes or no?

In regards to your concern, Google Scholar is not a valid profile for the verification process.

I didn't manage to obtain a clear answer about showing ownership of an institution-hosted webpage, but if even a Scholar account is not considered valid proof (even though it's the source of the knowledge panel), I assume that the institution-hosted webpage is certainly not a valid profile either.
Thus the answer is no, it is not possible for an academic to successfully claim their Google knowledge panel only with an institution-hosted webpage and a Google Scholar account.
Update: my claim was eventually accepted after I submitted my LinkedIn profile as single proof. Normally this is not sufficient according to the instructions, so I'm not sure what was the main reason for the acceptance. It's possible that me repeatedly pointing out the problem by email was a factor.