Before addressing this, a couple of points in your premise are flawed -
"...so help me God" is not part of the official swearing in. Someone can say that if they want, but it is not a required part of the official ceremony.
Before he enter on the Execution of his Office, he shall take the following Oath or Affirmation:—“I do solemnly swear (or affirm) that I will faithfully execute the Office of President of the United States, and will to the best of my Ability, preserve, protect and defend the Constitution of the United States.”
US Consitution - Article II, Section 1 - 8
The same goes with "God bless America" - that's a personal statement.
Now, on to your statement -
"in God We Trust" - of course it is a very specific religious statement, and intended to be a specifically Christian one. Yes, federal courts have ruled that it's some sort of generic, non-religious statement, but that's simply because SCOTUS decided they did not want to make what would be a spectacularly unpopular ruling to bar it, and basically started at their desired ruling and worked backwards from there. It was never part of the official lexicon of the US government at the time of the founding of the nation, and has only appeared and been added in response to upswings in religious sentiment.
During the Red Scare hysteria of post-WWII, early Cold War era, everyone was eager to demonstrate how non-communist they were, and the thing many fear-mongering conservatives loved to point to was the official atheist stance of the communist governments. That was added to show how non-commie and God-fearing we were, and by God-fearing they did not mean any kind of non-Protestant god. That's when it was officially, by joint Congressional proclamation, added to all currency.
“Nothing can be more certain than that our country was founded in a spiritual atmosphere and with a firm trust in God,” Bennett proclaimed on the House Floor. “While the sentiment of trust in God is universal and timeless, these particular four words ‘In God We Trust’ are indigenous to our country.” Furthermore, Bennett invoked the cold war struggle in arguing for the measure. “In these days when imperialistic and materialistic communism seeks to attack and destroy freedom, we should continually look for ways to strengthen the foundations of our freedom,” he said. Adding “In God We Trust” to currency, Bennett believed, would “serve as a constant reminder” that the nation’s political and economic fortunes were tied to its spiritual faith.
US House of Representatives History Art and Archives Historical Highlights: "In God We Trust"
Charles E Bennett, from Florida, was the member of the House of Representatives that introduced the joint resolution to enshrine "In God We Trust" as the national motto of the USA.
“In God We Trust” was first added to U.S. coins during the beginning of the Civil War, when religious sentiment was on an upswing and concerned Americans wanted the world to know what their country stood for. Many wrote to Secretary of the Treasury Salmon P. Chase on the matter, and he agreed with their arguments. Congress passed his act requesting the addition of “In God We Trust,” adapted from a lesser-known verse of Francis Scott Key’s “Star-Spangled Banner,” and the first two-cent coin with the phrase was minted in 1864.
By the turn of the century, however, the war’s memory had faded; President Teddy Roosevelt considered the mingling of God and Mammon to be vulgar, and he ordered the phrase removed from newly designed gold coins in 1907. A public outcry forced Congress to backtrack. By the mid-1950s, the concern with piety in Washington had apparently deepened; in 1955 Congress ordered the same phrase to appear on all paper currency.
....But as TIME wrote in that ’91 story, the banality of the phrases may not be worth the fight as a symbol of separating church from state. “Today even ardent separationists seem to agree with retired Supreme Court Justice William Brennan, who wrote in 1983 that slogans such as ‘In God We Trust’ have ‘lost any true religious significance.'”
Time Magazine: How 'In God We Trust' Got on the Currency in the First Place