Why don't U.S. elected officials swear on the Constitution, the supreme law of the land, instead of the Bible (or another "holy book"), in the spirit of true secularism and separation of church (religion) and state?
There are some elected officials who choose not to use a holy book.
Notably, John Quincy Adams recounted that he had used a volume of constitutional law, with the implication that he was swearing on the principles of constitutionalism itself. Franklin Pierce did not swear an oath at all, exercising the nuance of a constitutionally-mandated option to merely affirm the oath.
That this distinction was honored is, itself, important, because there actually is a difference. By definition, swearing is a divine invocation. This is a point of some religious significance to sects such as the Anabaptists, who foreswear the taking of oaths in compliance with the Sermon on the Mount. Keep in mind that the reason some of these sects put down roots in the New World in the first place was to escape religious disenfranchisement. So, owing to the same impulse as the First Amendment, Article II of the constitution recognizes that is not necessary to swear an oath, because it would be a point of religious qualification. Contrariwise, you might look to the example of Charles Bradlaugh, an English parliamentarian-elect who was found incapable of swearing an oath (and thus, qualifying for his office) because he didn't believe in the god that he was required to swear it to!
What I'm getting at, here, is that all of these people who have sworn oaths on the bible have done entirely so because it was their personal choice, not because any of it (even the swearing itself) is mandatory. In theory, anyway. Herbert Hoover and Richard Nixon were both Quakers, which would ostensibly direct them to affirm. Nonetheless, they both conformed to convention and chose to swear. Could this indicate a de-facto religious pressure on the office of President? Well, yeah, of course, but it's also possible that neither of those men placed much importance by the nuance. Times change.
Either way, if one does make the choice to swear in, it is only natural to substantiate it upon a holy text.
We may have Washington to blame.
For some of the prominence of such conventions, that is. You see, the actual content of the President's inaugural oath is constitutionally mandated:
Before he enters 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."
Wait, is something missing? As related by Washington Irvine, our first President apparently thought so. After he took the oath, he kissed the bible he'd sworn on and intoned "so help me god!" Concluding with such words remains a tradition to this day, though it is of course completely extraneous to that which is directed by law.
Because USA is not a secular country (e.g. the way France was). You seem to mistakenly equate/conflate the distinct concepts of:
"Freedom of religion" (which is what 1st amendment is about), meaning the government may not establish a preferred state religion or discriminate people for their faith.
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof...
"Separation of church and state" (a concept NOT found in the Constitution, and a much more recent legal invention - though based on writings of Jefferson, implying that government may not touch anything related to religion even if that would not rise anywhere to the level of "establishing a state religion").
Secularism, which is explicitly anti-religion-in-any-state-affairs. Frankly, I'm not sure there's a meaningful distinction between this and "Separation" concept, and would list them as the same thing.
Because the whole point of swearing is to swear on something holy to you (which is why non-Christians can swear on items aside from Bible).
The idea being, that you're making an oath in front of the deity. The implication being that breaking said oath is a sin.
However, the US Constitution is a legal document that does not have any holiness to it. There are no spiritual repercussions for not upholding an oath sworn on it, so you might as well swear on an empty piece of paper, or not swear at all.