Your question actually coincidentally intersects an area of my current research, so I luckily had some good sources at hand to shed at least a little bit of light on your question.

I want to first point out, however, that you inadvertently hit directly onto an inherent tension in the history behind your question: the stickiness of identifying any sort of essential character of the country. There is very much still a live argument, both popular and scholarly, over whether it is accurate to say the roots of this country are Christian, let alone deeply so. And at the time when the placement of Bibles in hotels (and myriad related practices) were most relevant, the legal and historical and legal-history communities were replete with arguments over just how Christian or secular the founding of and founder's intent for the country were. And I have two books in front of me at this very moment, both written by respected scholars, directly at odds over their answers to those questions, which I'll return to in a moment.

But first, let me get to your question more directly, and then I'll fill in some of the relevant context to understand the issues.

Gideons International is a non-profit, non-sectarian (meaning not specifically one Christian sect) organization formed in 1899 for the express purpose of helping to spread the Word of Christ (essentially, of both evangelizing Christianity and helping those who would like to do so themselves). The Gideons were founded when two traveling salesman found themselves sharing a hotel room in a very "no room at the inn" sort of scenario, which, ironically, I've never seen them or scholars note. I mean, if your founding happened in a situation very reminiscent of that surrounding the birth of your savior, you would think you might latch onto that. Or maybe they felt/feel it inappropriate, but I digress.

Either way, the Gideons are, for their first 40-some years, all traveling salesman: they would not let anyone else join the organization. Here I would draw a direct parallel to Alcoholics Anonymous, which was formed under extremely similar circumstances. Both were founded by evangelical Christians in the professional world (meaning having a profession like doctors, lawyers, businessmen) with a stated purpose for engendering Christian spiritual renewal. And while both relied for their membership largely on middle-class professionals, they both had an anti-professional (in the sense of experts) impulse: AA then as now eschews the use of doctors or other medical professionals in its services and the Gideons, though supporting and often coordinating with ministers, are lay businessmen (and their wives), though it's notable that as late 1937, only traveling salesman were allowed to join. These organizations all shared a concern for the spiritual health of men who, for much of Christianity's history, have been less religious on the whole than the other cis gender, even as they closely guarded their access to leadership.

But back to the Bibles, one of the earliest and by far the most recognizable missions the Gideons have had is provision of Bibles. They are most widely (and benignly) known for their provision of free, abridged Bibles to hotels and motels across the US. These abridged Bibles include the New Testament, as well as excerpts from Proverbs and Psalms from the Old. Now, believe it or not based on your question, this has pretty much never in any significant way been met with resistance: it's done by a private organization with the express consent of the private businesses and provided free of charge. The money is drawn from voluntary donations by members and supporters, and all the work is done by the quarter-million members worldwide. People can take the Bibles if they wish (even tear them up and throw them away, though the Gideons hope you will keep it and cherish it, of course), but they are under no obligation or even overt pressure to even look at them: they're tucked away in that little bedside drawer.

So the proximate answers to your question are these: virtually the entire reason they are so ubiquitous in hotel rooms is the Gideons. And the same can be said for why the practice is so universal: one very dedicated organization said, virtually literally, "We should put Bibles in every hotel room in the US." And it was partly to support men like themselves in maintaining a connection to their lord/religion, especially at first, but it was also to support evangelization more broadly. Indeed, today they are all over the world with a much more diverse pool of missions (a Christian word that can denote almost any kind of project related to services of the church, but especially those that help to minister to people). And no, there was almost no concerted effort to push back for the aforementioned reasons.

Now, one aspect I would like to touch on, and which deserves a much broader treatment, is that there are very close ties in the US between institutional Christianity (especially evangelical) and business. Evangelicalism especially places great emphasis on individual salvation, which is a quite miscible notion with the capitalist impulse toward particularization: individuals making private decisions in self-interest (with this word not necessarily having negative connotations with which we often associate it). Indeed, Kevin Kruse out of Princeton makes the argument in his book One Nation under God that conservative businessmen in the 1900s had a nationwide, concerted effort to sell the idea of the Christian nation, with the Christianity being one of personal responsibility, anti-collectivism, and individual liberty: essentially libertarianism and rugged individualism. And he is not really alone in this. Bethany Moreton's book To Serve God and Walmart (highly recommend) shows the connections upstart religious charities, schools, and other organizations have had to wealthy evangelical and evangelical-sympathizing business magnates. And while I do not have at hand any data on fundraising efforts undertaken by the Gideons, I can tell you that at least since there have been magnates (so roughly the late 1800s to now), they have played a key philanthropic role in the health and perpetuation of (especially evangelical) Christianity. Which is all to say that part of the Gideon's success is undoubtedly to do with broader trends making business and evangelical philanthropy good bedfellows.

That being said, the Gideons have been at the center or periphery of a much broader cultural battle that is at the meat of your question: the Bible in schools, which I'll follow with below (sources to come at the end).