There is concern when people of a certain religious faith wish to guide public policy in a direction that matches their beliefs, both on substantive policy grounds and a desire to separate church and state. It is a bit ironic, but not really, that we ban religious tests for public office (Art. VI and the First Amendment) while religion is such an important part of many who run for office. The barrier is not to a person promoting a certain religious viewpoint; it is in effect a guarantee that this will be in place for all comers. This doesn't mean it is always a great policy or that unofficially, the public decides certain faiths are more "proper" than others.
As noted in McDaniel v. Paty, there once was an effort to promote separation of church and state by preventing ministers (widely defined*) from running for office. This is no longer the policy, the outlier there declared unconstitutional as a violation of the First Amendment (one justice arguing it is a violation of voting rights). The right to equal access to public office also applies to the general believer, religious freedom based on individual belief, not merely associated with an established faith. It is left to the voter to decide who is best there.
The ruling was of the pre-Oregon v. Smith era, where equal application of general laws could still be a violation of the First Amendment, the old rule still statutorily protected under RFRA and allowed if state law requires it. The plurality here noted "free exercise of religion unquestionably encompasses the right to preach, proselyte, and perform other similar religious functions." This seems obvious to me. "Free exercise" cannot merely be "belief," but surely must entail other basic religious acts like going to services and so forth, though the core stuff often overlaps with speech and association (the "ministerial exemption" tends to involve religious associations having additional freedom of movement) in such a way that an independent right need not be focused upon.
Still, there seems to be something beyond belief and other First Amendment activities, noting that originally said activities might have been more limited than they are today (e.g., a focus on political assemblies; the Free Exercise Clause would add religious assemblies, leaving open various others that might be banned). Sacraments, for instance, involves meetings, words and acts. Thus, my concern regarding the ULC Church / marriage officiant controversies that religious freedom is hindered. Special favoritism or burdens on religion, especially certain religions, is particularly problematic. It also has arisen in the same sex marriage area, if we want to see a real religious freedom burden in place arising from that issue.
So, the proper approach is to accept that a Rick Santorum is allowed to run for office, but to be concerned with the merits of his plan for public policy. Also, "free exercise" involves various things, not just belief. The NY Court of Appeals in fact touched upon this when dealing with the contraceptives coverage issue a few years back (see here and my allusion here), the Supreme Court denying cert. NY took a half-way approach, providing limited heightened scrutiny to protect religious acts even respecting general applicable laws, such as one that might prevent use of ceremonial wine. Nonetheless, since the Church need not distribute contraceptives, non-believers are broadly affected and interests in sexual equality and health care arise etc. on balance made the requirement acceptable.
Anyway, Paty was a rather short oral argument, the state spending about seventeen minutes, might have spent less if a few questions added to his time.
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* The state ruling defined "priests" this way:
"1: one who performs sacrificial, ritualistic, mediatorial, interpretative, or ministerial functions esp. as an authorized or ordained religious functionary or official minister of a particular religion."; [and] "2. A minister in a non-Christian religion. 3. One whose role is considered comparable to that of a priest."A question in the oral arguments referenced a possible religion where each member is considered a "minister," another suggesting the Mormons so recognize. Quakers are specifically singled out in some marriage laws, e.g., since their "self-uniting" marriage ceremony has no officiant given the equal roles in that faith, the discussion there noting a federal court in at least one case suggested there was a federal right involved to equally application of marriage laws in such cases.