The history of Starfleet uniforms is long and varied, and reaches back earlier than the dawn of the Federation itself. But despite a wide-ranging colorful history, each era of Starfleet uniforms can be placed into one of two categories: lots of colors (one for each department), or few colors (departments grouped into divisions). The department-specific approach was used for about 80 years, from the mid 2270s (Star Trek: The Motion Picture) through the mid-century mark of the 24th century, giving each department (Command, Helm, Navigation, Engineering, Communications, Security, Services, Sciences, and Medical) a unique color.
For the rest of Starfleet history, a simplified structure has been employed, grouping various departments into what Memory Alpha and others have deemed “divisions,” with a single color per division. Despite many tweaks in the color assignments, the divisions themselves have remained remarkably consistent across 60 years of production history (and some three centuries of in-universe history.)
I propose that uniform color reflects fundamental differences in the basic approaches of each department.
Sciences
The departments in the “Sciences Division” focus on analysis of primary data sources.
The biologist works directly from her data, the doctor works directly from his patient's symptoms and readings. They are inherently skeptical and conservative in their approaches, methodical, with a preference for slower work that dives very deep. They synthesize conclusions based directly on their immediate findings.
Their work is generally fairly focused on a specific area– in theory, complications won't swing in from left field. Doctors have it more complicated– their approach does resemble those used in the command division, to some degree (see below). But the inherent skepticism, as well as the scientific rigor of their work and their historical connection to the sciences keep them within this division.
During the 2150’s, characters such as T’Pol, Sato, Cutler and Jessica Wolff all wore “Sciences Blue” at various points. From the 2230s through the 2250s, science officers on ships like the Shenzhou, Archimedes, and Discovery wore “Sciences Silver”, while their counterparts on Constitution-class vessels such as the Enterprise wore “Sciences Blue”. The Constitution variants were subsequently rolled out fleetwide, in a slightly different shade of blue. Roughly a century later, a variety of characters including physicians, psychologists and scientists all once again wear “Sciences Blue.”
The sole major exception to the color triad occurs during the first half of the 23rd century. Both on the USS Kelvin in 2233, and on the Shenzhou and on the Discovery during the 2250s, medical officers are shown wearing white. It is possible this practice was halted not longer after the Federation-Klingon War, perhaps specifically to avoid giving hostile intruders such obvious targets.
This division include general science officers, physicians, psychologists, astrobiologists, xenoanthropologists, and a range of other scientific disciplines.
Operations
The departments in the “Operations Division” focus on practical application.
They take scientific findings with which they are very familiar and put them to use; they problem-solve, with a low threshold for acceptable results: if it works (safely), then it's good. Engineers use physical scientific principles to problem-solve, while security and tactical officers use social science and strategy principles.
Creative, out-of-the-box solutions are encouraged, and being able to think on your feet and solve the problem in front of you is critical. Their work is also generally fairly localized, allowing them to focus very specifically on the problem at hand, before moving on to the next one.
During the 2150’s, characters such as Tucker and Reed wore “Operations Red”. From the 2230s through the 2250s, engineers on ships like the Shenzhou, Archimedes, and Discovery wore “Operations Copper”, while their counterparts on Constitution-class vessels such as the Enterprise wore “Operations Red”. The Constitution variants were subsequently rolled out fleetwide, with a transitional variant seen in 2265 which adopted the full-color tunic style, but maintained a beige coloring reminiscent of the previous copper (see note). Roughly a century later, engineers, security officers and tactical officers all wear “Operations Gold.”
This division includes engineers, technicians, operations managers, security personnel, communications officers, tactical officers (though see below), and sometimes certain administrative personnel including yeomen.
Command
The departments in the “Command Division” are required to engage in complex analyses and decision making.
They take information from multiple sources simultaneously and rapidly synthesize comprehensive conclusions, while potentially having to pivot their focus on short notice, temporarily leaving one problem unsolved, efficiently multitasking.
During the 2150’s, characters such as Archer and Mayweather wore “Command Gold”. In 2233, command personnel aboard the USS Kelvin wore “Command Blue” in a full-color tunic that resembled the Constitution variants shown some twenty years later. Also in the 2230s, then-Lieutenant Georgiou wears the “blue jumpsuit” variant, which uses a different color triad; given the precedent established in 2257 (and in the 2370s), it is likely that the Kelvin variants coexisted alongside the “blue jumpsuit” variants — perhaps the Kelvin was the Enterprise of its day.
From the 2230s through the 2250s, command personnel on ships like the Shenzhou, Archimedes, and Discovery wore “Command Gold” in the “blue jumpsuit” variant, while their counterparts on Constitution-class vessels such as the Enterprise wore “Command Gold” of various shades in the “full color tunic” variant. The Constitution variants were subsequently rolled out fleetwide, with a green wraparound variant seemingly only available to captains. Roughly a century later, starship captains and space station commanders, executive officers and flight control personnel all wear “Command Red.”
Aboard starships, the members of the Command Division are well-established: captains, first officers and flight control officers. However, as we peek further into Starfleet, we find other departments represented there as well.
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The fact that the oversight is so glaring is what makes me think that we are supposed to read in to the vague line about "sacrifice", and that we are supposed to infer that it's about Alexander. It's subtle and I think it's unsatisfying narratively, but I think that was the intent.
As to why they didn't just make an offhand reference like "Alexander is in command of an entire fleet"... I'm guessing they felt (rightly or wrongly) that it would be weird not to do more with the idea of "Worf as a father", so.... they opted to quietly make him Not A Father (Anymore). Which, to be clear, I'm not saying was a good decision, I'm just trying to do some tea leaves reading here.