Hi,
I’m not aware of any provision for applying for and maintaining a 60+ Oyster other than online. It’s a non-statutory concession so I think they’re free to operate the scheme however they feel is appropriate. You could contact TfL and ask them, or perhaps write to your London Assembly member as the concession is funded essentially by the mayor.
As a 60+ Oyster card holder myself, I wouldn’t be without it for any reason. You could set up a specific email address just for that purpose and only use it for maintenance of the 60+ card. It would be an interesting case study in proving whether or not TfL do share your data with anyone. I suspect that they don’t, but I have no proof either way.