It is essential for the full expression of religious freedom that believers be welcome, in law and in social custom, to bring their religiously based moral convictions into the ongoing public debate over how we ought to order our common life. Religiously informed moral argument does not establish religion or impose sectarian values on a pluralistic society.The statement largely equates religious freedom with being exempt from the ordinary, secular laws which apply to everyone else - you should be exempt, they think, as long as obeying those laws somehow violates your conscience.
But if you are religious you get to impose your specifically religious moral views on others if you can persuade the state to enact the appropriate laws - this is what is meant by the euphemistic phrasing: "bring their religiously based moral convictions into the ongoing public debate over how we ought to order our common life."
No, that is not essential for religious freedom; it should not be welcome in social custom; and it should be deterred as far as possible by constitutional arrangements that limit the power of the state to interfere with highly personal matters such as our sexual and reproductive lives - just the matters where religious believers so often want the state to exercise authoritarian control. The entire statement is written in a form of Orwellian Newspeak.
Notice how they ask specifically for exemptions from requirements on healthcare workers - requirements that have nothing to do with religious persecution, and would have been imposed on these workers if the relevant religions didn't even exist:
In the same spirit of concern for religious liberty, we ask that legislators formulate explicit conscience protections for health-care workers. And we counsel legislators to intervene and reverse the coercive efforts at the Department of Health and Human Services and other agencies to mandate health coverage and adoption procedures that will force religious institutions to betray their foundational principles. In these and other areas, we must vigilantly defend religious freedom.But no, this is not a breach of religious freedom.
Admittedly, it's a good idea for the state to lean against enacting laws that force people's consciences, whether their consciences are informed by religious ideas or by moral qualms based on secular ethics. However, the reason this is a good idea is that forcing people's consciences causes them suffering even if it is not religious persecution. Attempting to avoid that kind of suffering may be a reason for conscientious objection provisions in some circumstances, but if so there should be no discrimination between people whose conscientious objections are based on religion and people whose conscientious objections are based on secular ethical philosophies.
Furthermore, we should also not grant conscientious objections too lightly. Yes, you may suffer because your conscience is forced, but if the law is justifiable in the first place shouldn't we also be worried about undermining its operation? If exempting you will lead to harm to someone else - someone, let us say, whom the law was supposed to protect - what is the secular reason for the state to put your happiness first? And even if we do give you some kind of exemption, it may need to be narrowly crafted and confined - and you may just have to compromise if the exempting provision doesn't give you everything you want.
More generally, the underlying basis for religious freedom is the limited role of the state in protecting and promoting the secular interests of its citizens. In modern times, this might be extended to some of the secular interests of some people and conscious creatures that are non-citizens, since the state is at least competent to an extent to make judgments about those interests. Thus, we do have laws about cruelty to animals. However, the state is not competent to judge any creature's interests in such matters as spiritual salvation, and this, in turn, will give it a powerful reason not to suppress religions or to coerce people to abide by the tenets of a favoured religion.
This does not, however, mean that religious individuals and organisations get to pick and choose which legitimate secular laws they will follow. Prima facie, if a law is enacted for some legitimate secular reason, such as to protect the health of medical patients, and if that law has general application (i.e. it applies to everyone, irrespective of their religion), then everyone has an obligation to obey it. In those circumstances, you can't claim that you are being persecuted if you are simply expected to obey the same secular laws as everyone else. If the requirement you object to is work-related ... well, maybe you should take up some other career.
No one has a positive right to a job that they'll be able to carry out in accordance with their particular scruples. Nor do religious organisations have a positive right to run such things as adoption agencies and healthcare institutions however they want. If they don't like the regulations applying to these activities ... well, they are not being forced to involve themselves.