There are several factual errors in the video. One is, he stated that Jefferson was over the Senate and Trumbull was over the House (Speaker of the House) and that they approved the church services, but it was actually Theodore Sedgwick who raised the matter to the House on December 4, 1800, at the request of the House Chaplain, Rev. Thomas Lyell, a Methodist, and the Senate Chaplain, Dr. Thomas John Claggett, an Episcopalian, who were both appointed in November 1800. Jefferson was indeed President pro tempore of the Senate. However, according to the records of the Senate that same day and several days prior, nothing was mentioned about use of the Capitol building as a church. In fact, the Senate did not need to approve the matter since the request from both Chaplains came to the House for their Chamber. It is incorrect to say that Thomas Jefferson approved, in some official manner, church services in the Capitol.
Jefferson and others did indeed attend church at the Capitol Building, but that was largely because of logistics and an amazing lack of churches in Washington at the time, which was a really small community. Between 1800 and 1813 there were only three church organizations operating in Washington, only one of which had an actual church building. In 1825 there were 7 churches, all of them very small, all of which held services in their own buildings.
The Statement that Jefferson ordered the Marine Band to play at the services is one without any factual evidence, and is one of pure supposition based on the fact that in much later years the Marine Corps Band became known as the President's Own Marine Corps Band, and that notion was assumed to be from the start. The fact is the church sessions were orchestrated completely by the Chaplains and at least two books of the day indicate that the Chaplains, not Jefferson, directly requested the Band to play at the services. (Jefferson invited, not ordered, the Band to play at his inauguration, though, which started the tradition of if performing at every inauguration since, and eventually led to the President's Own moniker). It appears they played twice at the Capitol church services, on two consecutive Sundays. Both times they played traditional marches after the preaching ended, and the second time also played two pieces of psalmody which the churchgoers tried to sing along with, but it was characterized as a failure, and the Band was never invited back.
Church services in the Capitol were common throughout the 19th century. At one point, nearly 2,000 people assembled for church there each Sunday representing the largest Protestant church in the U.S. at the time. But as the community grew up around the Capitol the need for such services waned, as did the attendance. For 130 years there were no church services in the building until for the first time in all that time about 300 members of both parties in Congress gathered just hours before the passage of the health care bill. Congressman Randy Forbes (R-VA) led a prayer service on Sunday, March 21, 2010 in Statuary Hall, the old U.S. House chamber.
The people of Congress were generally religious people, and certainly approved of religion in the general sense. But there is no indication that the government of those people were endorsing or establishing a state religion, despite what people on either side of the issue may be. Thomas Jefferson attended a church service in the Capitol just two days after writing his famous "wall of separation between church and state" metaphor in a letter.