The very definition of "establish", in todays overly lawyered and litigious society, has morphed into meaning ANY mention of religion. Much less public funding for private schools.
Because the second section of the Blaine Amendment reiterated the First Amendment Establishment Clause verbatim (just repacing the reference to "Congress shall make no..." with "No State shall...") and because the underlying purpose of the Amendment was, as you say, to end taxpayer funding of Catholic parochial schools, its clear that "establish" meant something less than official sanction of a state church
Yet when you read the Congressional Record of the debate...congressman after congressman, both those who opposed it and those who supported it, stated that it would, for the first time, make the prohibitions of the First Amendment applicable to the states.
The most outrageous aspect of First Amendment law today is that it is all based on the 1947 Everson case...a case in which the Court held that the Establishment Clause (as incorporated against the states by the 14th Amendment) prohibited the state of NJ from indirectly funding catholic schools. What's outrageous about it is that the Court in Everson did what the Blaine Amendment tried to do. Of course the Blaine Amendment failed because, as you see from some of the quotes in post #32 in this thread, many Congressman objected to an amendment to the US Constitution that would have regulated their states.