I don't think anyone destroyed the base for the reasons not related to Cerberus. So far, I've seen only "they're evil," "I don't trust them," or "I blew it sky-high, lol, the look at TIM's face!"
There are plenty of people who destroyed the base for a simple, honest satisfaction of seeing TIM upset. "Who's the boss now?!" "Watch this, I quit!" "Oh how long have I waited for this moment" kind of reasoning. They don't like TIM, his authoritative and manipulative attitude, the way he's always in control of everything and doesn't share any of his long-term plans with you, the way he orders you around like you're one of his minions, as if you owe him your life, as if he OWNS you, and who cares that it's for the good of the galaxy when TIM behaves like he owns the place! Forget the Reapers - now is the time to show TIM who's in charge. It's not like the game gives you enough opportunities to do that. Extra cut-scenes with furious TIM!
I respect those people. They aren't trying to cover up their motives or justify them after the fact by some fake noble reason. They're simply enjoying the game. Extra content is always good.
Then again, in-game, they aren't fit to command. The galaxy would be doomed if the things were left to them; they'd pick some booze and some chicks and retreat to the edge of the galaxy to watch the mass destruction; when the Reapers finally get to them, they'd go down with a massive explosion, maybe even taking a Reaper or two with them, and kissing their LI with the unfolding brilliant sky in the background, too bright for the human eyes, before being consumed by a supernova. They like explosions, see. They're flashy.
Then there are people who honestly don't know what to do, but TIM's evil, so if he's upset by something, it must be good, right? And their teammates tell them what to do, too. All of the teammates disapprove of keeping the base afterwards. And the star is BLUE if you destroy the base.
This is settled, then. There's no need to think too much about it, once we figured out what the game wants from us, what it will punish and what it will reward. Besides, blue option always wins, right? There's no need for thinking over the choices. Just press that flashing blue button, and you're in for the gentle ride of win.
Then there are people who think that those who are called "terrorists" are the ultimate evil. There's nothing worse than being associated with people who are branded terrorists; the destruction of all advanced organic life in the galaxy doesn't even come close.
Never mind that Cerberus has never committed any acts of terrorism. The Alliance and the Council say that they are - it must be true. We aren't going to use our own brains to draw conclusions on our own. The government will tell us all we need to know. The alien government. Because such is the law. Cerberus is law-breakers. BAD. There's no need to think any further than this.
Never mind that a government, the morals, and the law are not a value in itself - they're simply instruments of coercion meant to protect the population and the society from destroying itself and collapsing; when the population is gone, the laws are meaningless, and the morals are useless. Would you like a "R.I.P. Milky Way, you were very law-abiding this cycle, we appreciate it" on the new gravestone in the Reapers' collection?
Besides, I've no doubts that the Reapers have their own government too, and they have laws that permit them the extinction of advanced organic life in the galaxy of Milky Way every cycle, and I'm one hundred percent sure that the Reapers declared Shepard a terrorist. So, are you going to stand down now, because you're acting against their law?
The alien government, namely Council, is not meant for humanity's benefit, but for the benefit of the three species in charge. And then some people would speak of "democracy." Wake up, what democracy? Three people rule the entire galaxy. The "lesser species" are not even represented.
When the Council makes a decision, they don't consult anyone. They don't answer to anyone. They just exchange glances and nods. The decisions they make might include: "gentle genocide" of your species, leaving your species to be wiped out by the unknowable alien force, moving their fleet here and there, starting or ending a interstellar war.
If you want your species' problems solved, you beg for an audience and wait humbly to be summoned, hoping that someone will pay attention to your plea. Maybe other species have no pride, no instinct of self-preservation - or maybe they lack the fleet, which is more likely. This is no democracy...it's the Empire.
And it's been working just fine for the aliens all that time, simply because it takes less time to make decisions when they're up to only three people. And the Spectres, the evil minions who follow the Emperors' every whim, sure are efficient at cutting the red tape.
Before that, it's been ONE person ruling the galaxy - the asari. If they'd been all right under the rule of asari for millennia, they'd be just fine under human dominance, too. We'll be benevolent to them - no worse than asari had been, anyway. They'd be even better under humanity than if they'd been left to the Council - they'd be alive.
The empire did a good job at giving a position in the Council to the species who become powerful enough to threaten it. Yet this position didn't give humanity any leverage. It's not enough. We can't even prepare the galaxy to the Reaper invasion. We need more power if we're to protect the galaxy in the future from the similar threats. The ultimate power would be sufficient.
The old Council proved their own incompetence. They nearly got the galaxy destroyed. They must be replaced by someone who can deal with the threat to galactic stability, and not ignore it. And I think humanity just fits the job description.
But the people who refuse to work with Cerberus would rather stand idly by and do nothing while trillions of people are being destroyed around us, because if we do anything, we'd be called "terrorists", and that would be BAD. We can't have that happen to our shiny Paragon image. Better let the galaxy die than have our reputation tainted. And advancement of our own species is BAD. And if humanity were to rule the galaxy, that would be just horrible. Worse that total extinction. Death can be reversed, but human domination is final.
Then there are people, the fearful ones, the "but what if!" kind of people.
What if Cerberus joins the Reapers? (not a single sign that they would, in all the previous games).
What if Cerberus fails the research, they're so incompetent? (We all learn from our mistakes, you know. The more experience you have, the less likely you're to repeat your mistakes. And if their research backfires, then Cerberus would be the first to pay the price and suffer the damage, so why do you care?)
What if Cerberus becomes indoctrinated? (No one is proofed against that, happens to the best of us, see the Arrival. You don't need a base for that, just a Reaper, and there are going to be plenty of them.)
What if they turn against you? The situation might become worse! Worse than the total extinction. It'd be, like, double extinction! We're going to die twice!
Then there's that argument, which is still all about Cerberus, "I'd have kept the base if I could give it to anyone but Cerberus. They're going to use it for evil!"
Whom, precisely, would you rather give the base to? The Council? You must really hate them, then. They're just going to declare that it's the geth tech. They'd wander in without protection and become indoctrinated. It would be a disaster. They don't even believe in Reapers, no reason to be cautious, and nothing you can say, do, or show them is going to convince them.
Maybe you'd rather give the base to the Alliance? They would just blow it up, because it's "an abomination" - whatever that means. And if they try to study it - see the Arrival. They've never dealt with such things before. They were never burnt by the fire. They'd put their bare hands right into it, just to check the temperature. They're going to build their research facility around a strange artefact. Cerberus' been through that. They're less likely to repeat their mistakes.
And even if the entire Cerberus becomes indoctrinated, they're still humans. You can kill humans with bullets, as Shepard proves again and again in ME3. It doesn't matter how empowered they are, they still go down when you shoot at them.
You can't kill the Reapers with bullets. We don't have a weapon to deal with them. We need something. Anything. We're not in a position to be picky. We're teetering on the brink of destruction here. We have only a few months left. Indoctrination of our first line of defense is acceptable price. The deaths of thousands is acceptable price. The tainting of Shepard's reputation is more than acceptable price - nobody dies, and Shepard suffers some isolation and a lot of hatred, but he'll endure. Their own tech, turned against them, would be just perfect. It's the thing that could save us. It's the only thing that truly "happens" in ME2 - we get our hands on their base. We can do nothing else to prepare to the invasion.
You don't even need to like Cerberus to preserve the base. Shepard sure doesn't - neither Paragon nor Renegade one. He issues a warning before he leaves - no matter what conversation options you choose. He's very hostile towards TIM. Doubtless he's already gazing into a divination sphere.
If Cerberus betrays us, it doesn't matter if they have the technology. We can still kill them. They're only organics - flesh and blood, easily destroyed. And intel leaks. Intel always leaks. Especially such highly volatile as Reaper tech. Cerberus will use it, and they will leave a trail, and they will be forced to consult someone else to unravel its secrets, and in this age of information, it will be all over the galaxy in no time. And it might be just enough to turn the tide of the war to our favour.
You will have to fight the indoctrinated anyway, whether you liked them and knew them as organics or not. You'll have to fight a lot of husks. The Reapers will turn plenty of organics against you. Indoctrination is their most insidious weapon, and you can't expect them not to use it. But before Cerberus goes down, if you preserved the tech, they can still save us.
Mind you, I'm relieved that even if Shepard blew it, Cerberus is still here to save the day and salvage some tech from the explosion. It's a good thing that the salvation of the universe does not depend on a single person, who can make grave mistakes like destroying the Collector base.