The National Anthem and the showing of the flag at sporting events were products of World War II and the Cold War which was an extension of it.
The addition of "under God" to the pledge of allegiance to the flag was added in the early 1950s at the urging of Billy Graham among others, since we were in opposition to the Godless Communists and this was a means of displaying our closer association with the Divinity. The pledge itself has an interesting history and one not divorced from the commercial nature of our society. It was not adopted by Congress until the early 1930s.
Prayers at public sporting events have been sometimes things in much of the nation, but more common in the South, and certainly more characteristic of institutions with a strong religious connection. I suspect, without the benefit of looking it up, that like the flag and National anthem, the praying over an athletic contest was not common in the nineteenth and early twentieth centuries.
These memorials represent culture and taste, and often a specific connection to a prevailing religious bias. As more and more people become less and less associated with that bias, the secular nature of our society will assert itself, as it has already done in some instances, and we may return to that earlier practice of seeing sport as entertainment, where prayer and patriotic displays will first be optional, then no more likely to be practiced than at other kinds of entertainment where they have never been much in evidence, if at all.
Religious institutions, competing against other religious institutions, may indeed be understood to have rituals that public institutions do not have, but when they play secular, public institutions, it could be thought that imposing that ritual in that sphere would be less appropriate.
One of the interesting, and often confusing, elements of our fast evolving culture is the practice of thinking of culture and style as moral issues, and generational conflict often revolves around this confusion. We all could think of multiple examples of this in our own lives and times, and since religion is inherently conservative, though conflicted in some aspects of this, resistance to cultural change and the evolving style which is a part of it seep into the pulpit and the pews, and that leads to inevitable generational skirmishing. We have all either experienced that ourselves or witnessed it in others.
The tendency to moralize on matters that are cultural is a natural one but is often a mistaken one, as the passing of time confirms. But religion by its institutional nature is less favorable to cultural change than the secular aspects of economic and technological movements which are ore permeable and make few claims to eternal verities, the arguments over the nature and scope of market economies aside.
This is an old theme in the history of our country and one that historians have properly taken as fertile areas of exploration over the last century of investigation.
Foreshadowed by observers from abroad, even before the notable Alexis de Tocqueville in 1831, this country has always been seen as diverse in its tastes and in its willingness to adapt itself to cultural changes both from outside our borders and from within. Resisting it is natural, but eventually accepting much of it is equally so.