As Becca says, I don't think privilege is something that can be done away with that easily or that is under the control of any of us as individuals (or of any individual relationships.)
Here are a few examples of what I think of as couple privilege (ie ways in which society privileges couples):
Invitations to events are almost always for a couple (if you are single your invitation will say "and guest")
Only couples can get married (in many places only heterosexual couples)
As an addendum to "only couples can get married" only couples can take advantage of the many, many privileges in our society only available to married folks.
Couples can hold hands in public without getting the fish-eye (again, in many places heterosexual couples only)
Couples (married or not) don't need to worry about being kicked out of their home due to zoning laws.
Couples (married or not) don't need to worry about losing their children just for the number of people in their relationship
Couples (married or not) don't need to worry about losing their jobs just for the number of people in their relationship
Every romance movie is written for/about couples
Every conventional (in the US) wedding ceremony is designed for couples
Couples won't be kicked out of their religious organization just for the number of people in their relationship
People ina couple are automatically assumed to be stabler, healthier, more reliable than single people of people in larger/networked relationships
I could go on. These are very real "couple privileges" that nothing any of us as individuals can do to change. It will take massive changes in society to undo them.
IMO, when the term "couple privilege" is applied to a specific relationship, what's really being discussed is one of two things:
either prioritizing the couple, which can be reasonable
IE:
My long term partner and I have 6 years together, are part of each others critical mental health support systems, have a kid and a home together, so sometimes I need to put this part of my life first--if it's a choice between the rent money and our date, I won't be able to afford our date. If my partner has a mental health crisis, I may need to be able to cancel our date to be there for him, because these are commitments I made before I ever met you. It doesn't mean I care for you any less, and if something in my relationship with my LTP does force me to cancel our time our will reschedule as soon as possible. I will do everything I can to meet your needs in this relationship. I hope you'll talk to me if you feel these prior commitments are damaging our relationship, and I will understand if you feel you can't be in a relationship with me because of my prior commitments.
Or Treating other people like shit and putting a fancy name on it
IE:
I am his wife, so I will always come first and you can only have time with him when I say you can. If you don't like it, there's the door.
Obviously (I hope) there is a lot of ground between these two extremes, and sometimes we all treat people like shit unintentionally. And sometimes you need to just talk shit out.
But (IMO) if you feel like you need a Secondary's Bill of Rights or a way to explain to people about how you deserve to be treated fairly whether or not you are the newer relationship, then those are probably not healthy people to be in a relationship with in the first place. It is not your responsibility to educate people who supposedly love you about "I am a human being and deserve to be treated like one."