I stand corrected. Am I misreading your explanation or did the Danish abandon strict linear tactics in favor of, to quote a handful of Civil War era tacticians, making the skirmish line the "main line of battle?"
Yes. Allready in 1848-50 did this happen.
When looking at period maps of the battles most units are shown as a square block with a lot of dots in front of it.
each "block" is a 200-250 man company with 2-3 platoons in column. and 1-2 (for a total of 4) out in front in "skyttekjæde".
(in the first war the army enlarged its size by making the companies 250 and later 275 men and later adding more units)
When the war broke out in 1848 the army already had a number of jäger battalions. and the line battalions also had men in each company with a (slow firing flintlock) rifle.
With 4 line battalion and one jägerkorps joining the rebels as I remember it.
In 1849 a number of line battalions was converted to light infantry battalions.
By the end of the war in 1850, about 25% of the danish infantry was armed with a rifle-musket and 80% of the "german" rebels had a rifle musket.
For most of the 1850ties all danish soldiers first learned to use a smoothbore, then the best 20% of the men in each company got a rifle musket also.
By 1860 the army started up rifling the smoothbores to minierifles.
The 1863 drill book have all the close ordered formations and evolutions one can expect.
But half the pages in the book is spend on how to fight in "skyttekjæde"
Where the closer order is strict and a lot of details on how to do thing, the open order chapters have very few clear instructions.
It actually make it a bit hard to understand, since it got fewer clear instructions with clear orders. And more a "get it to work" attitude.
In the battalion chapter, is It also have an interesting rule. It basically tell the battalion commander to let his company commanders do their wok without interference from him...
When this updated drill book was published the "secretary of war" Thestrup knew very well that a war was likely. (and against Prussia who had fast firing breechloaeded rifles)
Another difference is the fact that it include more about how you teach soldiers. That is language, attitude and similar and it also include more about when to do something, Where the american drill books only explain how.
Unfortunately many of the good ideas did not get used. The army more than doubles in size with mobilization in late 1863 /early 64.
By making each battalion into a regiment of 2 battalions. Resulting in a critical lack of officers and NCOs, too many too old soldiers and with the war starting in February and heavy snow there was no good way to do something about the situation.
So close ordered formations did get used way more than they should have had.
But usually with a platoon or max a company doing something in one formation. Not the long slow lines we see in north america.
And just to compare the two.
A american skirmish line by Hardee's is in one rank with 5 paces between each man.
A danish Skyttekjæde is in two ranks and with 4 paces between each file.
(both have options of making this more or less)
So the by the book danish version got more then 2 times the number of rifles pr. m of front line.
Another is how you do fire and movement.
In the US (french) system the two men of a file do fire and movement as a team.
In the danish system it is done as a unit. So if a platoon is in Skyttekjæde and need to move forward then you order 2 sections to fire and the other two to move.
And the ordered are just "forward" or "retreat" or similar.
With the use of whistles as an option is yelling is not sufficient.
Very much a 19th century case of "keep it simple stupid"
It even mention this in the first part. What ever you do, it need to work under combat conditions.