Here's an article listing what was left out, the big one being the mention of slaves being equal to 3/5's of a free person. [LINK - Washington Post]
Congressional leaders chose to read it as it is applied today not in its original, un-altered form which is fine with me. It's their coronation. They can have it their way.
My personal feeling is that republican leaders didn't want to tread on the view of many on the right, who see the Constitution with a religious-like conviction and see it as basically infalible. They accept it on faith that it is the correct, noble, unchanging law-of-the-land. To read sections that are easily seen as wrong or to give the idea that it could actually be changed is heretical.
But,.... Here's a very good post on why we should read the entire document, warts and all.
Here's a portion but you should read it all
Reading the entire Constitution is a way of reminding ourselves that the Constitution is always a work in progress; that it has been flawed in the past and probably is still flawed in the present; that what we have now before us is not necessarily the final version of the Constitution, but that the Constitution can always be improved and that it must be improved; that no matter how much our political institutions may have failed us in the past, and no matter how much we have failed ourselves in the past, political redemption is always still possible; and that We the People of the United States can still always strive for a more just, more free, and more equal country-- what the Preamble of the Constitution calls a "More Perfect Union."
Reading the entire Constitution-- including its oblique references to slavery--is a way of engaging in proper humility about the products of flawed human beings, but it is also a way of expressing faith in eventual improvement. If the Constitution once allowed great evils, and now it does not, perhaps someday we will be able to recognize the current evils it still allows, and ameliorate them as well.
Again, you should read it all.