I disagree; as far as I know the specific problems of global variables (over and above shared mutable state in general) are things that apply just as much to singletons. Things like absence of scoping, lack of clear ownership, and as you mentioned initialisation and shutdown, are just as much a problem for singleton objects as they are for non-object global variables.

Objects containing mutable state have some advantages over plain mutable variables (e.g. the object can enforce that particular invariants hold and invalid states are never made visible), but as far as I know those are just the generic advantages of OO encapsulation, and there's not really any specific advantage to encapsulating global variables in a singleton that doesn't equally apply to encapsulating a bunch of shared scoped variables into an object.