However, as co-equal branches of government, the House and Senate can set their own rules. Apparently there have been a majority who didn’t agree with Madison on this point.

You don’t seemed to understand Madison’s Constitutional argument. It is true that the House and the Senate can set their own rules, but they are not permitted to violate the Constitution of the United States in the process. As Madison noted:

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?

In other words, by enacting a law providing for the appointment of chaplains and their payment out of Treasury funds, Congress violated the First Amendment:

Congress shall make no law respecting an establishment of religion,

Had the two houses of Congress simply decided to open their daily proceedings with prayer, there would have been no reason to object on First Amendment grounds.

I think you can put your anti-religious minds at ease since from the subsequent history of both bodies that the impact exercise has been nugatory.

However that may be, one would think that a reversal of such an unlawful exercise would be equally nugatory. One would not have been able to deduce that, however, from the howls of outrage that greeted the 9th Circuit Court’s opinion a few of years ago that the words “under God” in the Pledge of Allegiance violate the First Amendment.

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 Congs 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 agst the members whose creeds & consciences forbid a participation in that of the majority. To say nothing of other sects, this is the case with that of Roman Catholics & 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 evil 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.
James Madison, Detached Memoranda

However, as co-equal branches of government, the House and Senate can set their own rules. Apparently there have been a majority who didn’t agree with Madison on this point. I think you can put your anti-religious minds at ease since from the subsequent history of both bodies that the impact exercise has been nugatory.

Only occasionally!

While we aren’t as badly afflicted by religion in politics as the USA, there still aren’t many openly atheist politicians here. Its nice to see one prepared to go on record against these archaic traditions.

Or, instead of wasting your breath on some prayer to an imaginary being, you could simply remind everyone at the beginning of the session of the importance of the decisions they are making. You know, cut out the middle-man.