I have a (completely made up out of nowhere) theory on that one. Basically the humans walked right in the middle of a political situation that was extremely lucky for them.
- The humans already built a small but still formidable military force in anticipation of first contact. Why they did that is unclear, either because Alliance leaders were paranoid or to stimulate the economy of earth and it's early colonies or whatever (in Revelation, Hackett speculates that Alliance crass already knew about another potentially hostile race being out there long before the First Contact War for example).
- During the First Contact War, humans show that they are very adaptive tacticians and a fairly aggressive and formidable species. That's how they first get on the councils radar.
- At this time, the council has issues with the batarians in the Attican Traverse. The batarians have been a constant source of trouble for the council ever since they showed upon the galactic map (see the second paragraph here). It is possible that in the 2150s and 60s they started another aggressive campaign in the region, maybe even reaching out to the Terminus systems.
- The council cannot afford another direct confrontation with the batarians because they fear that the conflict could escalate. Yet, they can't ignore their constant aggressive expansion that starts to summon unpleasant memories of the onset of the krogan rebellion.
- Enter the humans, who conveniently show up right at the batarians's doorstep. Because of the skirmishes during the First Contact War and the subsequent lsos of human life at the hands of a council species, they have the perfect excuse to jump start the human's development through "reparations", concessions, developmental aid, etc. in the next few years, all with the intent to foster a power struggle between the humans and the batarians in the Traverse and thus keep the batarians out of the council's hair (it would be just like the asari and salarians to cook up a plan like that, the turians will go along and the humans will be happy to be pawns in this game if it gives them advantages).
=> The humans develop like crazy, build one colony after another and even get an embassy in record time while the batarians are kept bottled up in the traverse.
It's a typical proxy war strategy, only in this case, the humans are the proxy and it all works to their advantage. It's only later, when the humans have their embassy and everything that the council tries to reduce their support (the discovery if illegal AI research by humans on Sidon gives them the opportunity to do so for example).
I also thought that humans were fortunate when they encountered the Council and I think there is plenty of evidence to support your Batarian theory, plus another conflict.
As I understand, the Council had ceased to expand for a period of centuries after the Rachni and Krogan wars. They appear to have made the same decisions as Hadrian and the Chinese had adopted after major crisis events. They had recognised their political expansion had progressed to a point where it stretched the ability to police and protect the Empire. That the costs of expansion were greater than the perceived benefits to the political elites. The became entrenched behind their borders.
They recognised Batarians into the Council after a century of economic trading between the Volus and Batarians. Batarian's became more aggressive in 1785CE and continued right through for centuries of skirmish wars; . Sometime around 1785 CE, a Batarian fleet bombarded the salarian colony world of Mannovai; in 1913, the Batarian Hegemony annexed the independent Asari colony of Esan; and in 2115, Citadel forces skirmished with Batarian forces on the planet Enael. The Batarians also adopted proxy policies of supporting criminal and terrorist organizations and a never halting slaving and piracy campaign, and expanding into the vacuum of space that the Council's entrenchment policy had produced.
I believe that the Council's escalated their entrenchment because of the Batarian crisis of the previous several centuries. They adopted a policy similar to the one adopted by Shogun age Japan where even the most minor transgression of the territories by "aliens" was met with extreme force. Then come the humans in 2157. I think you are right that Humanity may have been perceived as a way of the Council to tackle the Batarians at a greatly reduced cost, and limit the capability of both species, enabling both to be absorbed into the vassal states assigned to Volus, Elcor, Hanar; or -if the conflict between humans and Batarians weakened both species enough- ostracize them into the status assigned to Krogan and Quarians, or possibly anhiliated like the Rachni.
Pity for the Council and the Batarians that humanity, and the groups in galactic society that would benefit from a major economic expansion or increase in political influence allied and supporting the expansion- were able to be far more effective in the role that the Council Species Elites had assigned to them in the late 2150's-2160's. It was not a human expansion, but a multi species migration and pacification.
Some of the conflicts after 2160's were significantly larger scope than the ones in Dialog and were fought with colonial militia and PMC's. Probably most notably being Anhur Ebellions in 2176-2178. The Colony was founded in 2165 and has a reported population of 208,587,000.
A garden world with heavy populations of humans and batarians, Anhur was home to one of the ugliest violations of sapient rights in modern human history. A consortium of corporations and corrupt politicians, fearing batarian economic competition due to their custom of legal slavery, passed a resolution that abolished the minimum wage - effectively relegalizing slavery on a human-dominated world.
Opponents of the motion quickly turned to activism and violence. A civil war erupted as one side sought to end slavery throughout the system and the other, primarily a batarian faction called the Na'hesit, sought to keep the slaves they had. The Anhur Rebellions raged from 2176 to 2178. The Na'hesit had a significant advantage in ships, labor, and weapons, forcing the Anhur militias to hire mercenary companies to even the odds. In the end the abolitionists won out, though at the cost of much of their infrastructure. Though Anhur today still has significant natural wealth, it is economically depressed save for the reconstruction industry.
Nothing about the Alliance having anything to do with this conflict.