Getting to do something means taking an available opportunity to do something. It generally has a positive connotation.
get
verb (used without object)
21. to succeed, become enabled, or be permitted:
You get to meet a lot of interesting people.
Source: Definition of “get” on dictionary.com
When someone says “you don’t get to do that”, they mean to disallow that opportunity and suggest you are not able (or should not be able) to do something you want to do.
objectify
transitive verb
to treat (someone) as an object rather than as a person
She says beauty pageants objectify women.
Source: Definition of “objectify” on merriam-webster.com
The key part of this word’s meaning is that objectifying someone means not treating them as a person. Objectifying someone means relating to them as if they were anything but an independent consciousness, complete with a unique identity, thoughts, feelings, hopes, dreams, etc.
In the case of this headline, the author is suggesting that we have a tendency to relate to this infant as a phenomenon rather than a nascent being. Vawter seems the be saying that when we pin our own expectations on a famous baby we take away Blue Ivy’s freedom to define herself on her own terms as she grows up.