The Vatican does damn well in worrying about secularism in Europe as much as about Islam.
I think you are mostly correct. Secularism in Europe goes far beyond avoiding the "establishment of religion".
"In this sense, a struggle exists and so we must defend religious freedom against an ideology which is held up as if it were the only voice of rationality, when instead it is only an expression of a 'certain' rationialism."
European secularism is of the evangelical, sectarian variety. Holding moral values derived from religion has nothing to do with advocating a state religion.
What is prohibited in the U.S. Constitution is the establishment of a state religion. The "wall of separation" between church and state was coined by Jefferson in a letter to a Baptist who advocated a national day of fasting. Jefferson's "wall of separation" letter to the Danbury Baptists (scroll to bottom). Jefferson's letter is often overstated. Jefferson was referring to the federal government, only. Nor was Jefferson an absolutist. How could he be an absolutist? The two religion clauses of the First Amendment are in opposition to one another, and require constant balancing between competing interests.
Recently an Italian politician, Rocco Buttiglione, was ambushed by socialist EU politicians for professing mainstream Catholic beliefs. When the Roman Catholic Church must advocate for religious freedom, and has solid ground on which to stand, secularism has become a religion unto itself.
In the United States the "wall of separation" between church and state is secure. Courts have probably built the wall too high in many cases. Secularists are now crusading against any vestige of religious symbolism in the public square. The United States is not a "Christian nation". Neither is the United States a nation where Christian symbols or beliefs should be hunted to extinction within the public realm. It will always be a balancing act between the "free exercise" of religion while avoiding the "establishment" of religion. But when the 'great' battles involve nativity scenes on the lawns of public buildings, "under God" in the Pledge, and tiny crosses on flags, methinks the secularists doth protest too much.
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.
HOWEVER, radical Islam poses a threat to the socialist distopia EU politicians are trying to create, the Vatican, and basic human rights. I still maintain the jihad is a greater immediate existential threat to Europe. But the secular fundamentalists tend to side with radical Islam for its willingness to confront the United States using terror tactics. The cooperation between medieval theocrats and socialists never fails to amaze me. They are united in their hatred of the United States, and not much else.