Part of being an academic is being available to discuss your publications. Your full name will not only be flying around the internet but recorded permanently in libraries and journals.
Science is about collaboration, and standing behind the work you do, publicly. You will find it extremely difficult or impossible to get your PhD without being known to the academic community.
I think you won't find many anonymous scientific papers held in high regard.
I understand the concerns of privacy, but working in academia means that you give up some of the privacy.
Yes people will have your real name and they will know what college you work at and if some crazy person decides that they want to stalk you on campus because you're woke or part of the deep state turning the frogs gay with chemicals they'll be able to easily do that.
You're gonna have 100s of strangers in your classes during the year. You're going to tell them exactly when you're going to be in your office for office hours.
If you are unable to handle that I doubt academia is for you.
Academia is about furthering human knowledge especially a PhD. There are sacrifices involved; your privacy is probably one of them.