The 14th does that. It was put into place, because the States were violating the rights of their citizens. The 1st Amend has been incorporated, so that's no longer an issue.
" I couldn't find anything in the first amendment mentioning "public school system," let alone forbidding religion."
The public school system is a legal jurisdiction under the State Constitution and the US Constitution. The first Amend forbids establishment of religion and it protects freedom of expression. Freedom of expression doesn't apply to govm't entities, because that's expressly forbidden. From the 1st Amend, "shall make no law respecting an establishment of religion," That includes coercion by any school dist to learn it.
Most, if not all, state constitutions have a similar restriction.