Though that brings up the question: how would a conflict between, say, the 5th Amendment of the USC and perhaps a contradictory statute in one of the states be resolved? Because, until the 14th, it seems like you had no protections against any given state. And, arguably, to this day we have more interactions with state laws than federal.

If there were a direct conflict, then the U.S. Constitution would prevail under the Supremacy Clause. Technically, there can't ever be a conflict between the 5th and state law, because the 5th does not itself directly apply to the states - only the 14th does - but we can gloss over that.

I emphasize that the First Amendment didn't apply to the states because it's important in understanding what role it played in disestablishment. There were certainly Founders who were suspicious or hostile to the idea of state religions. But there were also Founders that were supporters of state religions (or the rights of states to have their own official religions), and who didn't want the new Federal government to be able to stomp all over those with its own national state religion. Some of the Founders were motivated by a desire to get government out of the church business altogether, but others were motivated by a more prosaic desire to separate the federal Congress from the state churches.