The argument is not based on a Constitutional imperative. Religions are numerous, contradictory, diverse and largely false. They are also ostensibly concerned with the supernatural. This implies two things; believers can make all manner of outrageous claims to special instructions from God which men outside of the faith are not privy to; and, more importantly, they can justify the extermination or conversion of nonbelievers. While it is presently true that most Christian religions in the U.S. are relatively benign it is the intolerant fundamentalist faiths that will aggressively dominate the secular institutions at the grass roots level.
Once this door is opened it is only a matter of time before the group with the most extreme and aggressive followers is in charge and anyone who doesn't toe the line will find themselves in a hell of these zealots' making. It is this possibility, however remote, that must be minimized by the simple expedient of keeping any form of religion from being sanctioned by an attachment to government institutions.
'tis why our Founding Fathers made a Constitution which said there shall be no established religion and no infringement upon the free exercise thereof. It is against the Constitution for the government to infringe upon my rights to exercise my religion as I believe. And since my religion goes with me wherever I am, i.e., God is always with my, that is my belief, I have the right to pray wherever I wish. That is not instituting a state church.