Not just a bone- there's a subtext that it's basically the equivalent of the Council ignoring the problem. Spectres are independent and without support? Sure. Right. And just so coincidentally, if the Alliance doesn't pick up the tab and do the searching themselves- like they would have to otherwise- then the Council can simply blame the Alliance and Human Specter.
Heck, the Council is even vague on punishing Saren if he is found and caught. There's concern, but the subtext with Garrus is that if you actually caught Saren alive, he might be deemed to valuable to loose and escape the noose. Hence why you can have a Renegade resolve to ensure it doesn't come to that.
Spectre status was always a weak joke in Mass Effect. It was supposed to put Spectres above the law, but it only applies where the Council already has primacy and can use the laws to it's will. We Spectres work almost exclusively where it has no pull- the Terminus, or the criminal underworld- and even where it does, the very thing that it's supposed to allow- public violence and breaking the laws- are still limiters. Tela Vasir still has to hide her involvement in a terrorist attack. Saren still loses his status on an accusation and weak evidence. Shepard still ends up in jail.
The best use of Spectre status wouldn't be for run-and-gun types like Shepard. The type of foes they go after are the ones where Specter status is irrelevant. No, the best specter would be a Volus who fights white collar crime, and uses the specter status to cut through red tape.
I'll point out, though, that ME:A could easily still have Specters if it wanted. Simply call them the sanctioned agents of the Arc Council- which, hey, offers us an easy way to have a Council leadership authority even without the Citadel. It was always the collaborative relationship of authorities behind the government, not the location, that made the Council the Council and Specters their agents.
Similarly, the Shadow Broker can still exist. Unless you exclusively define the Shadow Broker as either That Yhag or Liara T'soni, someone setting up their own spy ring and calling themselves the Shadow Broker is really all it ever took. The old broker's network went the way of the old galaxy- even staying in the Milky Way post-ME3 wouldn't keep it from being sundered with the relay- so a ground-up recreation would always be appropriate if it was going to continue.
And, of course, our own Council- composed as racial representatives of all the major Arc races, aligned for common cause in the galaxy but with differing needs and perspectives and whose authority is limited outside of those boundaries. The Arc Council certainly won't be the dominant force in the galaxy- but any rump council of post-ME3 wouldn't either with the collapse of the relays. Our Council being the little fish in a bigger pond is perfectly suitable and consistent with the themes of the Council when you recal that the Council hadn't even looked at 99% of their original galaxy. A Council wouldn't be the Council only if you persist that the Mass Effect Council must be defined as Asari/Turian/Salarian/Human and no other races (which, after the Reaper War, was almost certainly doomed to change- the Asari deceit, the rise of the Quarians/Geth, the Krogan- or that the Council has to be the dominant galactic power- which failed to be applied or relevant in ME2, and ended in ME3.
Depending on how the Arc comes about, we can have some familiar corporate names or influences as contributers to the Arc project if we wanted and thought it was that important. Founding backers like Binary-Helix, or security firms like Eclipse- who even now have offices and offer employment or services to members of the Arc with offices on the Arc and looking to expand in the new galaxy with knowledge taken from the old. These wouldn't be the offices back in the Milky Way- which we didn't exactly see or tour either- but these would be their offices and names all the same, changed for the changing context. If Exo-Geni isn't Exo-Geni because you define Exo-Geni as something that only existed in shop menues in the Milky Way, I'd raise an eyebrow at that definition.
The carrying of myths and culture, even if not organizations and governments, can allow for familiar things to find new life in a new context. Self-styled Asari Justicar vigilantes- because surely it's the idea of a perfect Code, and not the institution, they derive legitimacy from- challenged by context or the realities of moral imperfection. Self-styled STG, separated by who knows how many generations from the genuine STG, raising the questions of how a institution maintains legitimacy from an identity over time. A human cabal, hidden in the system where 'Human Interests' are in constant collaboration and cooperation, which holds a label that surpasses the longevity of the Nazis.
Many of these things can come over, if we wanted. Maybe we don't- maybe some things should end, whether they be left behind or allowed to die in the Milky Way. And maybe these things aren't so important after all to the Mass Effect experience. Eclipse was a mercenary firm will little lore and existed as people to be killed who primarily existed in one game. Are they really that important to Mass Effect, that Mass Effect was less of a Mass Effect because the first game didn't even have them at all?
Yes, things will change- but the nature or form of organizations was always going to change after the Reaper War, unless your problem wasn't merely the endings but the sheer fact that the Reaper War was big enough to fundamentally change the status quo of the political and economic structures of the Council System.
The real absurdity about Spectres when you think about it is the absolute erasure of sovereignty the council demands. Let's say Saren wants to take out a high ranking human diplomat. Can he just walk and blow his brains out and wave his Spectre card? Always having a troope of people within your borders who are all foreign nationals and completely immune from your laws seems to be a serious issue.