Name any other organization that was interested in recovering Sheppards body and spending whatever it took to revive him? The Alliance wouldnt do it for practical reasons not to include that they never believed that a whole fleet of Reapers was coming. Same for the Council.
Once Shepard was revived under Cerebrus hands practically nobody really trusted him. Even if Anderson and/or the Admiral used all their influence to give him another ship etc this would have taken time and per the evidence provided by Cerberus many other colonies would have died while he was waiting. Also what about the Intel provided by Cerberus? Alliance and Council would have never been able to provide this because they had no interest in investigating something they didnt believe was a threat.
Like it or not other than Cerberus and perhaps the Shadow Broker nobody had the resources and drive to make this Happen. You could ask for a different opening and having the council believe Shepard etc but that is not what happened and within what has been established Shepard had few if any options.
Regarding the Normandys upgrades, wasnt the new cannon of Turian make? If so I doubt that it would have been readily available for humans to use. Also the Collectors mission wasnt to go after Earth but just the colonies in the outer rim and perhaps if needed others closer to civilization. So they were a threat to the colonies and if left unchecked would have been able to produce a new Reaper. What doesnt make sense is why a new Reaper since the Reapers were around the corner anyway unless it had some strategic value by demoralizing humans.
But I don't buy the Alliance and the Council downplaying the Reaper threat in the first place. The writers knew that they had to justify Shepard working for Cerberus, so they forced those two groups into a characterization that made little sense in order to back Shepard into a corner.
Look at Anderson and the Council at the end of ME1. If you let the Council die, it's a human council. Why would they be ignoring human colonies? It makes no sense.
If you let the Council live, they clearly talk about the Reaper threat. Then you have the Normandy - one of the most advanced ships in the galaxy - get toasted by an unknown enemy and they blow it off like it's no big deal? Seriously, nothing about the characterization of those groups makes sense.
And in terms of trust, the Council gave me back my Spectre status. Once Shepard had that, I have little doubt that she could have gotten Alliance resources to help with this, since we already know that the Alliance was conducting military operations in the region. Why didn't they? Because you're actively working with Cerberus at the time.
Regardless, this is well-tred ground. If you bought into working with Cerberus, then more power to you.
To me, I'll never see it as anything other than writer fiat twisting and forcing the universe to fit into a new shape that doesn't really fit.