I was born in the UK in the 60s and I have a certificate of birth and a certified copy of an entry of birth both from the sub district of Sefton Park of Liverpool South district in the County Borough of Liverpool but my name does not appear in any of the UK archives and I currently live in a different country and have never personally used the birth certificate. Does this mean that I am not a registered citizen of the UK? Or is my birth certificate invalid? Why does my name not appear in my of the archive records?
If you have a certified copy of the register of births, then as the comments point out, by definition your birth was registered in the UK and a record of this should exist in the General Register Office (GRO) and local registry office archives. As records are not removed from the registry, then it should still be there.
I am not sure where you have been searching "the UK archives" but you can find information on searching the GRO records here. If you cannot find yourself in the index but you are in possession of a birth certificate, then the index is probably at fault. It is possible that the original handwritten index was entered into the system incorrectly - you may want to try different spellings?
(You wouldn't expect someone's name to appear in eg the National Archives records, so don't worry if you're not showing up there.)
Citizenship is a very different question - it is probable that you are entitled to UK citizenship, if you were born there in the 1960s, but I wouldn't want to say either way for sure. See the notes in comments.
[edited down to remove my confusion - thanks @phoog]