Lack of a uniform means nothing. Leia has no uniform. Han was a high-ranked officer in the Rebellion and had no uniform. That isn't evidence that Holdo's incompetent or just got her position for political reasons- its pretty much par for the course for major Rebel/Resistance characters in this franchise. So I feel like you're apply real-world military standards and logic in a place where they don't necessarily apply, and doing so inconsistently at that.
I do think that the top-heavy command structure that seems to be potentially in place could be due to drawing people from a wide range of backgrounds, some with existing high rank in other organizations and others needing to be appeased with a title.
As to whether Holdo being an NR officer would enable her to more effectively coordinate with other NR elements... that would depend on how much of the NR fleet is left intact, how much is still willing to continue fighting the FO, and how many contacts Holdo retained when she left, I'd imagine. If she was part of a fleet that largely surrendered/collaborated, and mutinied to defect to the Resistance, she might not have a lot of friends left beyond whoever she brought with her when she left the NR.
That said, while this is purely speculative, if I were writing her backstory I'd probably make her an officer in a local defense force, just because its more original than making her an NR officer.
But if we want more than just speculation... well, there's not a lot to go on, but this is basically all Wookiepedia gives on her bio pre-TLJ:
A human female, Amilyn Holdo was born on the planet Gatalenta during the early Imperial Era. In the year 3 BBY, Holdo joined the Apprentice Legislature, an Imperial organization on Coruscant for youth in politics. While in the Apprentice Legislature, Holdo met Princess Leia Organa of Alderaan, daughter of Senator Bail Organa. Holdo and Princess Organa spent much time together during senatorial sessions and routine pathfinding training on various worlds such as Alderaan, Eriadu, and Felucia. While pathfinding on the planet Pamarthe, Holdo discovered Organa's involvement with the rebellion against the Galactic Empire. Soon after, Holdo helped Organa find passage to the Paucris system in order to warn the fleet of an impending Imperial attack.[1]
Organa's Resistance
In 2 BBY, the rebellion became the Alliance to Restore the Republic,[6] and in the subsequent Galactic Civil War, the Empire was toppled and replaced with a New Republic.[7] However, an Imperial remnant reorganized into the First Order,[8] so in 28 ABY,[9] Organa created the Resistance to oppose them.[8] By 34 ABY,[2] during the conflict between the two groups, Holdo joined the Resistance military and served as a Vice Admiral and commanding officer of the cruiser Ninka.[3]
After the attack on Starkiller Base, she was posted on the Resistance ship Ninka. On D'Qar just before the First Order's arrival, she briefed the Resistance pilots from Cobalt and Crimson Squadron about the events of the annihilation of the Hosnian system as well as the subsequent destruction of Starkiller Base. She offered ammunition from her ship to help arm their squadrons. Having heard of Rose Tico's previous accomplishments with Cobalt Squadron, Holdo asked her to take a position on the Ninka as part of the maintenance team.[10]
Also, this:
"She's somebody who's a bit off-kilter, who sees the world through a prism most others don't understand. At first Leia thinks she's pleasant but weird, but as time goes on, it becomes apparent that there's much more to Holdo than you might guess when you first met her. We don't really have a lot of true oddballs in Star Wars, so it was fun to introduce one!"
―Claudia Gray, on Amilyn Holdo[src]
Amilyn Holdo is a character developed for the 2017 film Star Wars: Episode VIII The Last Jedi, in which she is portrayed by actress Laura Dern. The casting announcement was revealed on February 15, 2016,[13] and Holdo's name was revealed on May 24, 2017 in a Vanity Fair article.[14] During the Journey to Star Wars: The Last Jedi marketing campaign, Holdo appeared as a supporting character in the young-adult novel Leia, Princess of Alderaan by Claudia Gray,[1] and was briefly shown in the picture book A Leader Named Leia.[4] Holdo's appearance in Leia, Princess of Alderaan was received positively, with some readers comparing her to the character Luna Lovegood from Harry Potter.[15]
Which leaves a big blank space of over thirty five years that you can fill pretty much however suits your fancy. But it does indicate that:
a) Holdo and Leia were very old friends, who met during their youth in the Imperial Apprentice Legislature. Since I'm guessing that this would be a fairly exclusive group of youths, I would say that it is likely that Holdo is either very intelligent, from a wealthy/well-connected family, or both, and more likely than not from a family that was at least publicly pro-Imperial (though given she was friends with Leia and supported the Rebellion, perhaps privately anti-Imperial).
b) She's enough of an oddball that she's probably spent much of her life not being taken seriously, which might help explain her lack of patience for Poe's shit.
c) Holdo had some limited involvement in the early rebellion, but apparently in an espionage capacity, not a fleet combat or command capacity.
c) She was a late-comer to the Resistance, having only just joined when the shit hit the fan, and she was given command of a ship and her rank as Vice Admiral (simultaneously?). This implies that she did not bring any ships with her, and that she was given a very high rank right off the bat, relative to the size of the forces she commanded and her known combat experience. Indeed, were it not for the fact that Poe references her involvement in a prior battle in a seemingly positive light in TLJ, nepotism from Leia would be the obvious explanation to me for how she gained her command (although other explanations could certainly be devised, if one wished to be generous).
d) She was put in charge of briefing star fighter forces, and offered to supply them from her ship's stores (this suggests a certain generosity to her character that I wish we'd seen more of in the film, and the connection to the fighter squadrons helps explain her animosity toward Poe very nicely).
e) We have no idea what she was doing for thirty five years.
Now, none of this says "incompetent" to me, necessarily, and several things, both on-screen and in her backstory, suggest that's she's anything but incompetent, and that moreover she's a generally decent person and a loyal friend. But it does say that her personality is perhaps not best-suited to getting others to take her seriously as leader, and that her experience and skills based on available information lie more in politics and espionage than in fleet command. So its possible that she was posted to a position she was not best-suited for, perhaps due to a shortage of qualified officers, or perhaps due to Leia wanting to surround herself with people she trusted (I have a hard time seeing Leia putting incompetents in command due to personal favoritism, but I can certainly see her, in the wake of losing her son, husband/lover, and brother, being very, very cautious about who to place her trust in).
Incidentally, something vaguely similar was always my head canon for Admiral Dala, another widely-reviled female admiral who was supposed to be competent but was frequently seen as anything but- my favored theory was always that she had been a good ground commander who lacked experience as a fleet commander, and got posted to a position she was unsuited for due to nepotism from Tarkin.
Bringing it back to the original topic, all of this confirms my view that Holdo might very well do better if assigned to a more political or espionage-related position, though the evidence is still too thin to say for certain.