It's not so much as the game is saying no one is ever going to remember TIM as the guy who did help save the rest of the human colonies from being completely wiped out, it's just that he never was the kind of guy one should trust in the first place. That's who he is as a person. Every step of the way, from when you first find out about Cerberus in ME1 to finishing business in the Collector base in ME2, TIM, and ultimately Cerberus, was a questionable allegiance. From a mile away, I could see TIM was likely going to ask me something big at the end of our fight, and I knew I could never say yes to that kind of person. The only proper way of dealing with Reaper tech is blowing it the hell up, and TIM was a pretty reckless person with his research into Reaper tech. That you can remember from how he reacted to the Derelict Reaper and his other interactions in the books and comics.
And hey, maybe if you keep the base alive, Cerberus does pass on information to the Alliance that could even up the odds for the Galactic Resistance BEFORE Cerberus betrays everyone (Remember, we still have about 6 months or so before the Reapers arrive). Again, Bioware has said that they don't want every former decision from both ME games to feel as though you should start thinking 'Paragon=good, Renegade=bad, the world is black and white'. We'll just have to wait and see how the Collector Base will ultimately fit back in with ME3.
As for the Council, remember that they still are very suspicious of your loyalty to Cerberus, never mind your refusal to drop the "Reaper" theory they're trying to dismiss. So to have a possible traitor come up to you and ask that you come see this base s/he found in the galactic core that is proof of Reaper tech? The council would never buy it, they would definitely never risk their lives to go into such a dangerous area, and even if they did for a second, you would likely be returning to see Cerberus ships roaming the base that you didn't destroy. That would definitely make them think that this was all an ambush, and they'd likely try and kill Shepard, or at least brand him a criminal on the likes of Saren.
I just spent a whole game saying how he didn't control me, I was using him not working for him, I had no choice, I needed his resources saving the universe was more important and I didn't have to do it his way with no choice during the game to not do it like that. And now the game doesn't give us the choice to carry it on
Self-justifications aside, why would you WANT to carry it on? Like I said before, Shepard can't unite the galaxy to fight the Reapers if he's working with a pro-human terrorist group that even Ashley - our xenophobe in the making - wouldn't dare go near. Some things in the game have to happen in order for the plot to make sense. ME1 - Shepard has to track down Saren and stop Sovereign; ME2 - Shepard has to stop the Collectors; ME3 - Shepard has to stop the Reapers and anyone allying with them. Maybe he's aware of what's happened and doesn't like it himself, but TIM is working with the Reapers, and as their tool, we have to stop him and Cerberus. From here, I'm going to trust the writers to have that make sense.