The National Defense University Press published an outstanding article called Mercenaries and War: Understanding Private Armies Today (online: https://ndupress.ndu.edu/Media/News/Article/2031922/mercenaries-and-war-understanding-private-armies-today/ PDF: https://ndupress.ndu.edu/Portals/68/Documents/strat-monograph/mercenaries-and-war.pdf) that sheds an enormous amount of light on why mercenaries are effective, or at the very least why they have used throughout history:

• First and foremost, mercenaries were professional soldiers, unlike part-time citizen-soldiers or levies. "No longer trusting mercenaries, Machiavelli convinced the Florentine authorities to raise a militia instead, composed of citizen-soldiers whose loyalty to the republic would remain unflappable. But loyalty is a poor substitute for skill. These farmers-turned-soldiers were no match for professional troops, and the Florentines were soon crushed in 1512 by professionals. This military disaster resulted in the capitulation of the Florentine Republic, henceforward under papal control, and questions Machiavelli’s claims about the superiority of militias over mercenaries.". In more modern times, mercenaries have proven themselves in various theaters, such as Nigeria's hiring of private contractors to fight Boko Haram: "Recent years have seen major mercenary activity in Yemen, Nigeria, Ukraine, Syria, and Iraq. Many of these for-profit warriors outclass local militaries, and a few can even stand up to America’s most elite forces, as the battle in Syria shows."
• Secondly, and to more directly counter the arguments against them, mercenaries could be and often were actually cheaper than standing armies: "The reason is simple: Renting force is cheaper than owning it. Maintaining a permanent military seems normal today, but it is not. Paying for one’s own armed forces is ruinously expensive, similar to owning a private jet versus buying a plane ticket when you need it. Why invest in your own expensive standing army when you could just rent one?" Mercenaries allow you to save money by only paying for troops when you actually need them, as opposed to standing armies which by definition are maintained indefinitely. Entering the 17th century, mercenaries were key in enabling states to keep up with the growing scale of conflicts like the Thirty Years War: "Clever military enterprisers outfitted whole regiments and leased them to those in need of martial services—the first military industrial complex...Rental regiments allowed rulers to wage war on an industrial scale without long-term administrative costs, like taking care of wounded veterans or pensions, and this lowered the barrier to entry in war while encouraging ever-larger battles." Also notice how the some of current hottest regions for mercenary action mentioned by the article (countries in Africa and the Middle East like Nigeria, Iraq, and Yemen) are poor with without high-quality standing armies. In combination with the first point, mercenaries allow poor nations to hire forces that are both cheaper and more effective than their standing armies: "That’s when the Nigerian government secretly turned to mercenaries to fight Boko Haram...they drove out Boko Haram in a few weeks. The Nigerian military could not achieve this task in 6 years."
• Thirdly, mercenaries could supplement existing armies, enabling campaigns or victories that wouldn't be possible without them: "Nearly half of William the Conqueror’s army in the 11th century was made up of hired swords, as he could not afford a large standing army and there were not enough nobles and knights to accomplish the Norman conquest of England." To put it another way, he was able to double the size of his force through the hiring of mercenaries. Darius hired Greek mercenaries to supplement his Persian troops, while Alexander the Great brought along 5,000 mercenaries in his invasion of Asia. Julius Caesar used Germanic mercenaries (specifically as mounted troops) alongside his professional Roman legions; this last example also shows how mercenaries can be used to not just "fill out" an army in size but round it out in capability by providing a troop type that may not have been available in significant numbers otherwise. With the native Roman citizens recruited as the iconic sword-wielding Roman legionaries, troops like cavalry and archers would have been mostly provided by non-citizen auxiliaries, allied contingents, or mercenaries.
• Fourthly, even though mercenaries are loyal to money and not the countries they fight for, this is actually a good thing for nonstate actors such as wealthy aristocratic families, religious organizations, and cities like Florence. Since nonstate actors don't have territory or citizens to recruit from, they need to find soldiers elsewhere, and mercenaries have continued to fulfill that need even in modern times: private military contractors have been used by charities and NGOs to protect their humanitarian operations in dangerous regions, while those desperate for pay have been hired by terrorists. Oil companies can hire PMCs to protect their wells, and billionaire Igor Kolomoisky used them to protect his assets by seizing UkrTransNafta's HQ (an oil company): https://www.vox.com/2015/3/23/8279397/kolomoisky-oligarch-ukraine-militia. As the NDUP article simply states: "Without mercenaries, nonstate actors had no way to challenge state ascendancy."
• Fifth, the loyalty to money rather than the nation can also make them useful in civil wars and revolts since they have no stake in country-specific conflicts and thus little reason to betray their clients: "King Henry II of England engaged mercenaries to suppress the great rebellion of 1171–1174, because their loyalty lay with their paymaster rather than with the ideals of the revolt." The disloyalty of mercenaries is also somewhat overstated, as solid client-merc relationships could be built up over years of trusted service and payment: "Sir John Hawkwood was one of the greatest mercenary captains of the age. Hailing from England and a veteran of the Hundred Years’ War, he was monogamous to Florence for over two decades, until his death in 1394. The city honored his faithful service with a funerary monument in Florence’s famed cathedral."
• Sixth, while there is always a "buyer beware" aspect of hired troops, the ability of mercenaries to betray their clients also works the other way around and provides several opportunities that national troops don't:"Buyers have ample opportunity to swindle mercenaries. Marketplace stratagems include: bribing the enemy’s mercenaries to defect, retaining all mercenaries in the area to deny the enemy a defense, and reneging on paying mercenaries once they complete the military campaign. Sometimes, clients hired a larger mercenary group on a short-term contract to chase off or kill unpaid mercenaries." That's not all: clients can also send mercenaries into high-risk engagements to ensure high casualties, making them too weak to demand payment afterward. Clients may also send multiple mercenary units after the same objective and only reward the one that accomplishes it first. Mercenary units can also be hired to fight each other, further weakening their ability to betray you or demand payment.
• Seventh, mercenaries are also useful for intrigue: because anyone can hire them, they can be deployed with plausible deniability. "Similarly, clients might hire the private sector for “false flag” operations—for instance, secretly hire mercenaries to instigate a war between one’s enemies, reducing them while keeping the client’s name out of it. Alternately, one can hire mercenaries for mimicry operations to frame enemies for massacres, terrorism, and other atrocities that provoke a backlash." In terms of public relations and diplomacy, they can also afford to be a bit brutal with your enemies than your own soldiers: "Some clients might prefer to outsource human rights violations rather than have their troops caught in the act."
• Eighth, mercenaries may be better for wars of attrition. Being a business that can recruit it's "employees" from anywhere ("Professional men of arms filled their ranks, coming from different countries and united by a paycheck..."), they can theoretically sustain conflicts longer than national armies that rely on the citizens within their own territory: "The mercenary labor pool is global, allowing longer wars of attrition."
• Ninth, while there is somewhat of a modern stigma surrounding the idea of fighting for pay, mercenaries have always been a fact of war (even appearing in the Bible) and that the historical line between national soldiers and mercenaries is a little bit blurrier than one might think. Military service didn't become linked to nationalism and patriotism the way we think of it today (i.e. "serving your country" by joining up) until the Napoleonic Wars, while money has always been a motive for fighting whether it's loot and ransom on the ancient/medieval battlefield or enlistment bonuses and pensions on the modern one. "People view soldiers like wives and mercenaries as prostitutes, who turn love into a transaction. But every soldier has a little mercenary in him, and vice versa. Troops often reenlist for big bonuses, a transactional practice common in most militaries." On the other side of the coin, mercenaries are not so...mercenary in their acceptance of contracts as you might think: "The author has also seen mercenaries refuse jobs on political grounds. Some American-hired guns will never take money from Russia, China, Iran, or a terrorist group; America’s enemies are their enemies."