The Founders in their wisdom and exactness of language as to intent added that little word ‘neither’ before adding the language/words you quote. As such in my thinking the first stated wording as to requiring a ‘natural born citizen’ was not abrogated in any way as being a specific requirement but was reinforced intentionally by providing that in addition to such the person must also have ‘..attained to the age of thirty five years, and been fourteen Years a resident within the United States’.
If you are saying that they are separate and independent requirements — that one must be a natural born citizen and additionally must meet the age and residency requirements — then I think you are correct.