However, all wallet software is centralized (whether open source or proprietary).
Is bitcoin core centralized?
If we are talking about software, then I think yes. It's currently running on my local PC which is pretty centralized.

Of course, it is not fully functional in offline mode, but it still works, even if only partially.
That's not what centralized means. When you run a full node on your PC you become a "peer", in other words you don't have authority over anybody or anything, you are just an equal to everyone else running a full node.
Being centralized means there is a single central authority making decisions like in case of services such as exchanges where the owner makes all the decisions.