I love Hugo Black; he was the greatest Justice the Court has had in all the years since he was appointed, but Engel has been one of the decisions that I have been suspicious about since I was a junior in high school.
If it is unconstitutional for public school teachers to lead voluntary prayer, then why isn't it unconstitutional for Congress and all the state legislatures to hire chaplains to lead voluntary prayer in legislative chambers? And why isn't it unconstitutional for military bases and prisons to hire chaplains to lead voluntary prayer? The Court itself had a Court Cryer who would always announce, when the Justices entered the courtroom, "God save the United States and this honorable Court."
As you already know, I share your appreciation of Justice Black. The most obvious answer to that would be different Courts, but that's obviously not a sufficient answer. Other than that, school is quite different from the other examples.
As for Congress itself, I would say that it is probably beyond the jurisdiction of any court, including the Supreme Court. The Constitution does a good job of insulating and protecting Congress from the other branches. (Or at least that's probably the best answer I can come up right now.) If Congress is drawing from the Treasury to pay for the prayer, that would certainly be unconstitutional. When it comes to state legislatures, I'm not sure, but I would err on the side that any payment would violate the First Amendment. As far as the Court itself, I'd be curious to know what the individual Justices thought back then. I think standing would be a big issue there.
When it comes to military bases or prisons though, that's something else altogether. I think that's where the Establishment Clause hits the Free Exercise Clause head-on. Neither being in the military nor being in prison ends one's rights under the First Amendment, though they can be limited (particularly when incarcerated). I would argue that there could or should be reasonable accommodations made for religious services under certain circumstances.