We're not talking about a doomsday weapon here. We're talking about deliberatly destroying advanced tech just because you don't like the receiver and what he might do with it.
If the receiver is someone with his own agenda that won't necessarily align with the good of the galaxy as a whole, it's reasonable to conclude that keeping it may cause more problems than it solves. The base isn't a doomsday weapon, but the one thing we know it *can* do is kill people and use their biological material to create a new Reaper. Maybe it could be adapated for defensive uses as well, but maybe not. Cerberus has had its own experiments get out of control more than once, and their leader has misled and manipulated Shepard before and has obvious ambitions beyond just stopping the Reapers. Miranda, who knows TIM best and probably has a better idea what he's likely to do with it, openly objects to the idea of keeping the base.
If I'd had the option to give it to someone I actually trusted, that would be one thing. If I could do whatever I wanted with it, I'd have Shepard go back through the relay with Mordin, Garrus, Tali, and Legion (as the most science/technology-oriented crew members) to see what they can salvage, and assuming they don't find a risk of mass indoctrination, I'd use it to try to get the Council's heads out of the sand and work with R&D teams from the Alliance and any other reliable species-based militaries to see if new defenses can be reverse-engineered from the base's technology. But the Illusive Man is just too much of a wild card - it would be like handing it to the Shadow Broker or Aria.