Absolutely. Because if they chose to not say any other part of the oath the same outcome would happen. Again, it wasn't the statement precluding them, it was their choice not to say it.
Let's look at the Oath
Each person enlisting in an armed force shall take the following oath: I, XXXXXXXXXX, do solemnly swear (or affirm) that I will support and defend the Constitution of the United States against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; and that I will obey the orders of the President of the United States and the orders of the officers appointed over me, according to regulations and the Uniform Code of Military Justice.
Everything in this refers to the duty of those serving. Everything references something essential.
Here's the rest of it and what is being objected to
Anything in the first section refers to the duty of the individual and references the authorities to which he or she answers to, but the second does not. The first is essential, the last superfluous and therefore not equivalent. It serves no purpose relevant to the "mission statement" or however you choose to view it.
So let's explore "choice". Here you go. I say the Oath, but the regs say it isn't complete until I take off my clothes and let others piss on me.
Well, it doesn't violate any of the oath. It isn't unconstitutional, in fact there's less of an argument against it than the conclusion of the oath. How about rape? Beating? I mean technically there's no Amendment saying you can't. Oh yeah the Code might have something to say about those, but you aren't in the armed forces until you complete your required ceremony. Oh, yes it's illegal, but not violating the law is your choice. It's not the process, it's your unwillingness. Your fault you know. I'd have a problem with that too.
The intent was never to exclude religion from the public square.
I agree.
They did not wan to promote or deny any religion. That was mainly the reason that people moved here to begin with and setup the United States, to get away from the church in England.
I bit more than just getting away from the Church. The CoE was a government creation which had many many powers that effectively made not so much a religious entity as a government organ. God, King and Country were all one in the same. That's what the Second was intended to prevent. Our churches can't tax or caused to be taxed for example.
As I said I believe that the Founders never intended for the practice of religion to be curtailed or forbidden as it has by some. You aren't dealing with a rabid atheist intent on removing all references. Still it comes back, at least for me, to what the boundaries are for government. Not just what you can't do, but what it can legitimately tell you that you MUST do, and I think that last bit ought to be limited. The Bill of Rights wasn't put there as an absolute minimum test. It's an explicit limitation on what some things the government cannot interfere with and it is most certainly not a comprehensive list. "Rights retained by the People" If all the rights one has is in the Bill, then we were screwed and quite intentionally from the beginning. No, I don't believe that the Constitution was designed to allow the government that kind of power. As noted by others the tradition has been to allow a choice here, something the government has removed for no good reason. The oath stands as it always has, but the power of the government to coerce has been increased for no possible positive reason. No, I don't like it.
You really aren't arguing for anything other than an extension of government power to force one to do what they believe is a significant breech of liberty to meet a condition of employment which is completely irrelevant to the task. I think Uncle Sam has too much of the wrong kind of power, why would I want to increase that?