Chaplains in the military and Congress prove that there is no separation of church and state.
Public funding of chaplains in the military, Congress, and other areas like prisons and the police creates a challenge for church/state separation. The existence of government chaplains doesn't prove that church and state are not or should not be separated, especially if these government chaplains should not exist. Opponents of church/state separation who base their arguments on the existence of government chaplains rarely do much to explain why the state should pay for chaplains at all.
The rationale for military and congressional chaplains is that when the government calls people away from home to render services to their nation, it takes them away from their home churches and religious community, a potential infringement on their right to free exercise of their religion. Doesn't it make sense, then, for the government to make up for that by providing some means of religious support?
The fact that government sponsored chaplains existed from the beginning is not a sign that the Constitution's authors never intended to separate church and state,
James Madison wrote in Detached Memoranda:
Is the appointment of Chaplains to the two Houses of Congress consistent with the Constitution, and with the pure principle of religious freedom? In strictness the answer on both points must be in the negative. The Constitution of the U.S. forbids everything like an establishment of a national religion. The law appointing Chaplains establishes a religious worship for the national representatives, to be performed by Ministers of religion, elected by a majority of them; and these are to be paid out of the national taxes. Does not this involve the principle of a national establishment, applicable to a provision for a religious worship for the Constituent as well as of the representative Body, approved by the majority, and conducted by Ministers of religion paid by the entire nation.
The establishment of the chaplainship to Congress is a palpable violation of equal rights, as well as of Constitutional principles: The tenets of the chaplains elected [by the majority] shut the door of worship against the members whose creeds and consciences forbid a participation in that of the majority. To say nothing of other sects, this is the case with that of Roman Catholics and Quakers who have always had members in one or both of the Legislative branches. Could a Catholic clergyman ever hope to be appointed a Chaplain? To say that his religious principles are obnoxious or that his sect is small, is to lift the veil at once and exhibit in its naked deformity the doctrine that religious truth is to be tested by numbers, or that the major sects have a right to govern the minor.
If Religion consist in voluntary acts of individuals, singly, or voluntarily associated, and it be proper that public functionaries, as well as their Constituents should discharge their religious duties, let them like their Constituents, do so at their own expense. How small a contribution from each member of Congress would suffice for the purpose? How just would it be in its principle? How noble in its exemplary sacrifice to the genius of the Constitution; and the divine right of conscience? Why should the expense of a religious worship be allowed for the Legislature, be paid by the public, more than that for the Executive or judiciary branch of the Government. ...
Better also to disarm in the same way, the precedent of Chaplainships for the army and navy, than erect them into a political authority in matters of religion. The object of this establishment is seducing; the motive to it is laudable. But is it not safer to adhere to a right principle, and trust to its consequences, than confide in the reasoning however specious in favor of a wrong one.
None of this mattered when the Supreme Court decided the constitutionality of legislative chaplains in Marsh v. Chambers, 1983. According to Chief Justice Warren Burger's majority opinion, chaplains are constitutional simply because they have always been around — they are now a tradition. There was never any attempt to judge the practice of hiring congressional chaplains against tests, like the Lemon Test.
This was probably because chaplains would immediately fail any sincere legal tests. Chaplains have absolutely no secular purpose. Second, they have the effect of advancing certain religious beliefs and inhibiting others. Finally, the existence of chaplains clearly results in an entanglement of religion and government. Even the earlier Sherbert Test would be violated, because there is no "compelling state interest" in having taxpayers support the existence of such chaplains.
Government-funded chaplains are, in almost all cases, wrong. First, their appointment and support easily leads to charges of religious discrimination and religious divisiveness. When the government becomes involved with recognizing certain religions or religious beliefs in an "official" capacity, there will be demands that other religions or beliefs be "officially" excluded. This is inexcusable because the government has no authority to show official or financial preference to any particular religious beliefs.
Second, the appointment of official government religious leaders can be used for furthering politics rather than pastoral counseling. This is just an example of the "golden rule" — whoever has the gold, makes the rules. When the government is paying the bills for religious leaders, the government can and will decide what those religious leaders are allowed to do and to say.
All of this is why the separation of church and state exists in the first place — and these are the reasons which were first articulated by James Madison. Government should not interfere with religion or religious faith (i.e., by telling priests what they may and may not preach) and churches should not interfere with government (i.e., by creating a preference for certain religious views and restricting or discriminating against others).