I will wager an attempt at providing a satisfying response, but I cannot do more than scratch the surface of all the developments that occurred in 1850-1914. And that is in fact, the gist of my answer to your question:

Is the change in standard arms responsible for such change, or were other factors present and effective during such a period?

No, it is not solely responsible (as usual). It is a small portion of the whole, but even on its own it was still a very significant development. The development of more effective rifles was just one development among many during this period. These include (but are definitely not limited to) the discovery and widespread implementation of:

• Smokeless gunpowder (1863)

• Practical repeating rifles

• Practical high explosives such as TNT

• Machine guns

• Telegraph communication

• Mechanization, mostly logistical (of which the train networks are perhaps most notable).

• The first 'modern' artillery, using advances in metallurgy and recoil systems, to get unprecedented range and payload sizes.

Again, this is by no means exhaustive. There were many more technological developments as well as societal changes and changes in military doctrine. Each of the developments changed the battlefield in a significant way; the improvements made in hand-held small arms is not to be underestimated either. Smokeless gunpowder did not just prevent a shooter from giving away his position, but also cleared the battlefield of the enormous smoke clouds that had greatly influenced battlefield tactics in the years before. Repeating rifles increased the firepower of common infantry many times. Moltke mentions this in 1858 already:

"The firepower of an infantry platoon [today] surpasses the range and destructive effect of the case-shot of a six-pounder cannon."

^(Note that firepower generally does not refer to the total mass of projectiles fired, but rather a combination of effects, such as lethality, area of effect, rate of fire, and more)

The improvements in rifle technology alone thus understandably altered the battlefield rather significantly already. But any actually observed changes must be seen in the greater context, where they were accompanied by other, equal -or even larger- technological developments. I would recommend reading "The Machines of War" by Hacker, and more generally: reading up on how modern artillery works compared to Napoleonic era artillery, and how the changes in capabilities affected the artillery doctrines. The effects, that each of the points I mentioned had, on the battlefield are really far too large to describe in any detail in a single post, and culminate into the WW1-style battles that you may be more familiar with.

^(As a sidenote I will mention, that it is essential to learn about how battles were fought on the Eastern front as well, when looking at the various effects of technological development, as the Western front is not entirely representative.)

To conclude with your main question:

How impactful was the shift from muzzle loaded weapons to full metal cartridges in the second half of the 19th century?

It multiplied the firepower of the individual rifleman manyfold, which alone certainly made a significant impact on the battlefield. There were however, many other important developments during that same period, all of which together resulted in the radically different warfare in the resulting World War.

Sources:

Moltke's tactical problems from 1858-1882

Barton C. Hacker; The Machines of War

Edit: grammar, conclusion

The simplest answer for the colorful uniforms was the use of black powder. In addition to being dirty and difficult to clean which prevented the universal adoption of rifles until the invention of the minie ball, black powder also released an enormous cloud of white, puffy smoke with every shot.

When you consider that every shot fired by every solider in every war up until the invention of smokeless powder (late 19th century) was highlighted on the battlefield by a huge cloud, certain things start to make a little more sense. On humid days or days without a lot of air movement or rain, clouds would linger and build up, and after a volley or two a battalion might have maybe ten yards of visibility in front of them. Rifles won't hit what a rifleman can't see, and ditto for artillery or other heavy weapons.

Colorful uniforms, far from being the result of a lack of imagination or commanders eschewing stealth, were considered far more important to keep an eye on battalions in the field, helped to prevent friendly fire, and were useful in showing off - an extremely important psychological edge on the battlefield.

American forces didn't adopt smokeless powder until 1893, when they adopted the Springfield Krag-Jorgensen bolt-action rifle, but in the Spanish-American War, quite a few national guard regiments and volunteers were equipped with the .45-70 trapdoor rifle, which still used black powder. Even so, the Army tried to move away from the bright blue wool uniforms for something a little more durable and rugged in the heat of the Philippines, and it was there that the somewhat ubiquitous khaki uniform came into service, at least for the US, in fits and starts throughout the conflict in the Philippines.

There's no one factor, but the adoption of smokeless powder and its lack of the telltale white puff of smoke was one of the elements that caused armies to mute their colors a bit.

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A fabulous overview of the US army pattern of build-up, disbandment, and re-building is in Gregory Urwin, The United States Infantry, An Illustrated History 1775-1918. He has an extensive section talking about the hodgepodge of uniform replacement going on in the Philippines.

Richard Holmes' Redcoat is another great book for dispelling some of the commonly held beliefs about the style of linear warfare.

I want to mention that when WW1 began people expected a short mobile war and the germans began with a large offensive through belgium towards Paris. During this period of the war trenches were rare and the war was relativly mobile with the french and british retreating towards Paris and the Germans following them. It was not until the british and the french halted the german offense and drove them back over the Aisne where the germans dug in with trenches and barbed wire. The britsh could not take their position and decided to dig in themself. At this point the front became less mobile. I also want to mention that not all uniforms were muted, the french still had their bright blue uniforms, as seen in this picture

Source:https://www.worldcat.org/title/world-war-one-a-short-history/oclc/695008475&referer=brief_results Note: I used a norwegian translation of the book, but could not find it, so i linked to the original