It hasn't stopped other southern and only slightly less religious states not having the law anymore. e.g. Louisiana at 73%. So I guess it's possible.
As far as I can see, Louisiana is not a case of not having such a law 'any more' - they never did. Their constitution has been modified many times, so I might be missing it, but I can see no sign that religious belief was ever used to qualify for office. Things which have historically disqualified Louisianans from holding public office:
- Fighting or serving as a second in a duel with deadly weapons
- Serving in the confederate armed forces
- Being black
- Being poor
- Being a clergyman or religiouis teacher
Nothing about atheists, though.
By way of analogy, it's still a matter of statute law in the UK that you can legally shoot Welshmen with a bow and arrow under certain conditions. Obviously, you can’t actually do this, since the laws are unenforceable under the legal doctrine of implied repeal. More recent homicide legislation overrules these regulations, but they have never been formally repealed and so still remain on the books as quaint historical monuments. If Parliament were to sit down and explicitly repeal such legislation, I think you'd be justified in condemning them as wasting time in an already full legislative calendar by repealing laws already unenforceable.
With these US State Consitutions, repealing provisions would presumably be a lot more difficult than a simple legislative measure. If the state’s rules require referenda for constitutional amendments, then it would be not only time-consuming, but expensive. Is it really worth spending a lot of public money on a measure which can only be symbolic, even if it succeeds?