Several reasons actually.
1) The best intelligence service/ the best spy is the one nobody knows exists or works in the area. The spy you know for a certain is a spy of the foreign country reporting on interesting developments is called "part of the embassy staff" or maybe "the ambassador" and the place those types gather is called "the embassy". Starfleet Intelligence is a known intelligence service whose agents you can identify. How do you identify somebody as an agent of opposing force if you do not know that opposing force even exists?
2) The Federation has been consistently shown to be a federation of goody two shoes, optimistic, diplomatic, altruistic, democratic idealists. In fact, it makes a lot of sense "the people" of the Federation would want their democratically elected government to have very close supervision of what is called "instruments of force", military, intelligence and law enforcement. Military in this case means Starfleet, intelligence services are Starfleet Intelligence and law enforcement is left to local police of every member, with Starfleet occasionally assisting the locals, if the miscreant has spaceships or has ran away off-world (see Bajoran Militia / Odo's security personnel for an example). Starfleet personnel and Federation ruling council have consistently been shown to be unable to compromise their ideals even in dire situations (TNG "Picard and Borg drone Hugh" incident could be taken as an early example, the whole "Voyager" series started because of one of those, but there are many more examples of that, both in TOS and afterwards).
Ruthlessness during intelligence operations and utopian idealism in which "Every species is potential friend and we don't kill people without proper justice proceedings, unless in self defense." do not mix well. So intelligence services will suffer in effectiveness. If an agent kills enemy spy during a mission, you can be damn sure there will need to be a report justifying it. If an agent destroys data or commits sabotage, report. If an agent commits multiple murders for the good of the Federation, report and report and a dozen other reports to make sure those were absolutely necessary and it was really for the greater good. And that's for asking forgiveness after the fact. Asking for permission to do that... yeah, you're not going to get that permission. Thus, Section 31.
A bunch of fundamentalist, Knight Templar type fanatics, willing to get their hands dirty and bloody to ensure the safety of the Federation. And while they each know they personally do not live to the "possibly too good to be true" ideals of the Federation, they are certain they do what needs to be done, so those ideals can even be given a chance in this cold cruel universe. Everything that needs to be done. No deed is too low, no price is too high, for the safety and continued existence of the Federation. Even genocide of the species, to remove the future threat. And that way Federation is not the hypocrite. Because they genuinely don't know it is happening. And can always say: "Oh, those were just some misguided fanatics. Sorry about that. They thought we will like that. We obviously don't like that. We didn't know they even exist. Them doing bad things in our name... Goodness gracious, NO! Absolutely not!"
And react the same way as US government would react to actions of a terrorist group inspired by US constitution. (21st amendment to be precise. The one repealing 18th, prohibition of production and sales of alcohol)
So why does Starfleet Intelligence not do that, just remove the oversight? Because they are part of the Starfleet and Federation and it would go against ideals of the Federation and would be considered a power grab. And Federation is above such things as power grabs and money and greed and all those bad base parts of human nature. Right?