Conscientious objection is a complicated area - I don’t think it’s possible to make broad statements like these. (Not to mention that if conscientious objection is built into the law, it’s not violating the law.)
In the United States, for the most recent wars, conscientious objectors were given the opportunity to server in other ways - in WWI, those who refused to do non-combat military work were jailed, but since then non military options have opened up as well, because we realized that that was wrong. I see no reason to work backwards, especially in this case where an 1) accommodation could easily be reached without harming any of what the government has decided are its interests, and 2) there is no worry of Mrs. Davis faking her objections out of cowardice or similar, since she gains nothing material and in fact has already been imprisoned.
Again, broad statements are difficult, but I’m reasonably confident in saying that conscientious objection does not need consequences of legal punishment when a) by claiming objector status, the person gains little to nothing outside of being able to follow their conscious (eg, safety from not being in a military), and b) there is little to no burden on the government or anyone else by accommodating the objection. In general, substituting some other duty makes sense, as is done in the case of the military, but even that doesn’t seem applicable here (though if it turns out that it is, I don’t think anyone would object).
In the case of Davis, the licenses could simply be issued under someone else’s authority. All that need be done is for the state to say that deputy clerks can issue licenses in their own right, or similar, and we’d be done.
Hardly a matter worth imprisoning someone over.