Having had a quick glance over the wikipedia article, I noticed that the assertion (Lee, 20 regiments, etc) has no source attached to it at all. The subsection seems to have been plagiarized wholesale from a book on early modern weapons, which also fails to give a source. I can't seem to locate any other sources for the contention, which makes me very suspicious of it.

Now, with that said, pikes did exist during the Civil War, primarily because they were cheap and easy to produce. It's an understudied topic, but that's in large part because they were so militarily insignificant that few historians bother to even mention them. I can't think of a battle in which they were even present, much less played a decisive role. The pike seems to have been favored more by rhetoricians and revolutionaries than by fighting soldiers.

John Brown, the abolitionist guerrilla of Bleeding Kansas fame, brought a large store of "pikes" (which were really Bowie knives on poles) south with him to Harper's Ferry. He planned to use these weapons, along with the muskets in the Arsenal, to arm a slave rebellion which he would then lead on a war of liberation. Revolutionary rhetoric aside, it's not hard to understand why he chose pikes rather than, say, Sharps rifles, when you compare the prices: $1 per pike, versus $30 or $40 for the rifle.

Governor Joseph Brown of Georgia also involved himself with the pike. In February 1862, with the Confederacy in crisis and threatened at multiple points, he saw the pike as a way around the logjam of arms shortages. As such, he called upon Georgians to build pikes and Bowie knives. And while some of these were actually produced, they saw very little, if any, combat use.

As to why the Confederacy never embraced the pike? Absent a smoking gun document buried in an archive somewhere, all I've got to offer you are some educated guesses. First, the pike, as you said, was an obsolete weapon on a mid-19th century battlefield. Second, the Confederacy was intensely image-conscious and sought to appear as a modern and developed country ready to join the community of nations, especially to European eyes; having battalions of pikemen running around isn't especially conducive to that. Third, Civil War soldiers tended to complain bitterly to their relatives, neighbors, and politicians when they were issued old-fashioned or poorly maintained muskets; it's not hard to imagine the effect pikes would have. Finally, it just wasn't necessary. While there was a period early in the war where the Confederacy had more troops on hand than arms to equip them, this became less of a problem as the war wound on.

I find it interesting that pulled were considered obsolete - a bayonet charge seems like it'd differ little in utility from the same charge by pikemen.