If you read far enough down, you finally get to the Napoleonic tie-in. What he's really talking about is skirmishers and "light infantry" tactics, and he's generally correct in what he's saying - he's just saying it in a way to make it appear different/controversial. I don't think it really is; most people that have studied this know that ACW tactics were essentially Napoleonic.
Having said that, my impression is that the tactics in use in the US would probably best be described as very similar to British Rifle unit tactics in Spain during the Napoleonic Wars. By 1861 many, if not most regiments were equipped with rifle muskets so these tactics make sense. I don't have the impression (and here I'm less well versed than in Napoleonic and 18th Century formations) than either side used anything that looked like the French assault column during the ACW, even for bayonet charges.
Burke seems to be advancing the idea in his posts that ACW regiments seldom fought in firing lines and I don't think this is anymore the case than is the misconception that the Napoleonic French never deployed in line. There certainly would have been plenty of tactical situations where the firing line was the best formation for the job - but by the 1860s, this was not the three- to four-deep, tightly spaced formation of the 18th and early 19th centuries, but an open order formation much more closely related to the British infantry tactics during the American War of Independence (see "By Zeal and Bayonets Only" for an excellent, in-depth discussion of the shift in British tactics away from European linear formations to open order, two-rank formations and skirmishers).
In most films and even written accounts about the war, attention is typically focused on the grand assaults, like the Union attacks at Fredricksburg or the Pickett-Pettigrew-Trimble charge at Gettysburg. While I don't know that skirmishers weren't deployed by the attackers in either of these cases, I would suspect that given the intent of these as assaults, their use was limited. Reenactors seldom engage in skirmishing either; rigid firing lines pouring volleys into one another shows better (and makes more smoke) than skirmishing does.
Skirmishers almost certainly were used extensively later in the war (and Burke alludes to this, though he seems to argue that they were as extensively used early in the war) and I can't imagine they weren't deployed during the Wilderness campaign, for example. Burke may well be right about the use of clouds of skirmishers early in the war for the same reason they became a staple of the French Revolutionary Wars: the amateur nature of the armies. Troops that aren't well drilled will tend to disperse into a "cloud" as described.
The other reason that ACW armies almost certainly used a lot of skirmishers is the same reason the British army switched to open-order and light infantry tactics in the Americas during the AWI: the terrain in the US during these eras was not at all conducive to more rigid, linear tactics. The open fields of Gettysburg between Seminary and Cemetery Ridges are somewhat unusual for ACW battlefields. Other major battlefields are covered with fences, orchards, copses of trees, outright forests, and other obstacles. Nothing short of an open-order formation and skirmishers has any real hope of maneuvering in that terrain. Even large parts of Gettysburg are really not great for deploying a line and trying to advance it; anyone that's visited Little Round Top and given it any thought can see that there's no way you're keeping a regiment in a linear formation as you advance up that hill.
In terms of gaming, about the only ACW board game I've played in the past decade is Great Campaigns of the American Civil War, which isn't of a scale to worry about skirmishers. The John Tiller ACW games, as well as Matrix's "Brother Versus Brother,: The Drawing of the Sword" both allow skirmishers to be deployed and doing so has a noticeable impact on firefights. As one might expect, it's best to recover them before launching an assault, which generally reflects the points made in Burke's series of Twitter posts.
I can't comment on games from GMT's Great Battles of the American Civil War because I've never played any of them or even read the rules. Comments I've seen about them suggest there's enough depth there that I would think skirmishing is somehow represented. Many other games covering the period seem to be more grand tactical in scale, and skirmishers are abstracted away.
Like Napoleonic gaming, ACW gaming really needs to capture skirmishing and light infantry tactics better. Jim covered this for Napoleonic gaming in our joint talk on the subject last year at Origins. Burke may have slightly overstated his case, but he's not at all wrong, and if you're going to have a tactical ACW game system, it really does need to capture this aspect of the tactics of the period to represent things at all accurately.