If the influence of organized religion was diminished because of increased secularization, what would America be like?
I think that the question makes the presumption that organized religion dominates American public and private life. This foundation could be easily questioned. A strong case could be made that increase secularization is already present. American political and social institutions recognize the division between spiritual identity and secularist forms of the good. Mainstream political and social notions of the good are steeped in a form of secularist identity.
Secularism is evident in how social and political institutions maintain a significant distance from organized religion. In American society, there is not "one organized religion" that dominates all facets of life. It is in this light where one can see the presence of increased secularization. I think that greater analysis and detail would be needed to see the America of today as one where organized religion dominates and controls so much of life. In my mind, increased secularization has already been realized and embedded as a part of American social and political identity:
While vigorous re-shaped forms of national public religion exist that utilize churches and religious authorities, they speak with a curiously ambiguous and ambivalent voice. They appeal to secular values and eschew much of the "God talk" that defined the social forms and doctrines characteristic of confessional religion in the early modern period. To some observers (e.g., Carter, 1993) this ambivalence is prima face evidence of the more general decline or evisceration of public religion in America.
One can see a potential rise in the role of religion in the lives of American individuals. A Pew Group study in 2013 found a rise in Americans who profess some type of organized religious identity:
The new, nationwide survey by the Pew Research Center’s Forum on Religion & Public Life asked Americans whether having “more people who are not religious” is a good thing, a bad thing, or doesn’t matter for American society. Many more say it is bad than good (48% versus 11%). But about four-in-ten (39%) say it does not make much difference.
While this might emerge from the study, Americans are not inclined to assert that there needs to be a wholesale transformation of organized religion in society. Part of this arises from the pluralist mode of spiritual exploration that is present in American society. The divergence in spiritual exploration in American modes of religious experience causes individuals to perhaps admit its need in the private realm. However, recognizing this and its different forms compels a more secularist embrace in social and political realms. It is in this light where one might see a rise in spiritual affiliation with organized religion. Yet, given how there cannot be one dominant spiritually explorative voice in American society, secularism is embraced to ensure that one mode of organized religion does not take away from another.
Secularization cannot be avoided. It must be worked around by careful preparation of the church. It is doubtful that the increase of organized religion can be directly associated with the increase of secularism. In fact, it may be close to a tie.
When religion becomes crystallized into organizational structures, people and the groups they were supposedly made for can become forgotten or smashed by the weight of the structures one struggles to maintain basing the methods on secular approaches which everyone adopts.
If you talk, for example, about evangelical Christianity, you will mention how important people are and you will have programs which adapt to each homogeneous group or context, but if your personal fellowship level does not grow in depth and extent, you end up living just to maintain schedules and fantastic programs which become more and more like the non-separate or non-religious activities.
Community sense and belonging, however, have become the theme of both secular and religious missions for many groups. Therefore, the line can be drawn at certain sections but not taken in the full gamut of what can be understood as the organized church.
Even though the case is not always moral, for example, as when immoral ideologies take over the messages, there has to be a concern to make clear distinction as to what is the church and what would be a club or any other organization or gathering.
Now, churches should grow in all aspects except away from the sound doctrine. And America has suffered from the kind of secularized patterns that have always destroyed families and societies throughout history. Here we are talking about the relativism which finds the breaches at the churches walls and which fails to imagine de necessary division between what is secular and what is holy. As to whether its religious organizations have decreased because of secularism itself, it is questionable. It may be that we increase our external efforts when we wish to avoid the needed changes inside.
As far as Christianity is concerned, we have from the New Testament's recent knowledge that under persecution, the spiritual Church, the Body of Christ increased with dispersion only as more families were reached at the ends of the earth.