You’re missing the point. That third party (which at least you acknowledge must exist) MUST have a monopoly on law & retaliatory force.
If it doesn’t, you don’t have a third party. You have numerous parties (i.e. various gangs) warring against one another using violence to impose their whims.
The need to prevent that and establish a system that allows individuals to freely exercise their rights is the rationale or reason for such an authority
I suggest reading Chaos Theory by Robert Murphy or at least this chapter of David Friedman's book, The Machinery of Freedom so we are on the same page. Government services aren't magically immune to economic laws, and instead magnify the failings inherent in trying to disregard them. Security and adjudication are not fundamentally different from other services.
Government doesn't "allow individuals to freely exercise their rights," instead infringing upon those rights as a necessary and fundamental aspectbofnits operatuon, and this does not provide "rationale or reason for such an authority." Governments are various gangs warring against one another and against us.