I am very much of the opinion that nation-states and the systems of government they use are two separate things. Systems of government come and go - after all, the one the U.S. has now is radically different from the one it had in 1789, so much so that if all of the changes had occurred all at once, you’d consider it a revolution.
That presumes that independence is something that is granted rather than something that is taken. The Treaty of Paris recognized U.S. independence - it didn’t grant it.
Glancing at the various independence days listed in Wikipedia, the majority are based on declarations of independence with a mix of treaties and ends of occupation thrown in.
My mother’s birthday was July 4th. When she was a little kid, she thought all the parades and fireworks were for her birthday. Even though it’s posthumous, and even though she had reality all sorted out by the time she was 7, don’t take this away.