I really wish it did, but the constitution does not contain that phrase. It does contain "Congress shall make no law respecting an establishment of religion" (the establishment clause) but "separation of church and state" comes from a letter written by Thomas Jefferson.
Except the US constitution does not include that language. The "a wall of separation between church and state" phrase most notably comes from an 1802 letter by Thomas Jefferson to the Danbury Baptist Association. Not a legally binding document by any means.
I imagine you're thinking of the Establishment Clause of the Constitution which forbids the US government restricting the free exercise of religion.
I believe, iirc, the Supreme Court over several decades has affirmed and reaffirmed the overall position that the US government must remain secular and not favor a particular religion. Which is effectively what you're getting at.
You are correct. Though the meaning behind the phrase remains true and biding precedent. If only precedents meant anything in today's court system. They reinterpret the Constitution on a daily basis to suite the billionaire needs whose paying them that day.