If everybody were allowed conscience exemptions from ordinary general laws, then exemptions would make sense—but we might as well give up any pretense to the rule of law.
When people are only allowed exemptions to ordinary general laws for “religious” reasons, then either we’re already halfway down the road to giving up the pretense of the rule of law, or we are nakedly giving some people special privileges because they present their moral judgments, and express their moral sentiments, in “religious” terms.
Maybe giving some people that special privilege of exemption is a way of ensuring that only “religious” people have the full array of choices as citizens, while people whose moral sentiment and judgment operate on other terms have some lesser standing within the law.
Or maybe allowing the special privilege of exemption from ordinary general laws disincentivizes the holders of that privilege from engaging, in their capacity as citizens, with the process of shaping public policy. Why bother to express your preferences in the public sphere when you have the privilege of enacting them individually, and without legal consequence, on your own?
There might be some middle ground, by which reasonable exemptions may be granted for reasons that do not require classifying people according to the terms and methods of their morality, and so that exemptions are rare enough that nobody worries about how they affect the integrity of the rule of law. But I have never heard an argument for such a system. Until I do, my lot stays with the opponents of exemptions.