That breaking up a monopoly (and how do you define one, anyway?) is good is not obvious (economies of scale, etc). But even taking this as given, that sometimes monopolies are bad, it is far from clear whether allowing government to intervene will be better.
You have to weigh the costs and benefits. Anti-trust law is extremely subjective. It's existence incentivizes companies to use it against each other to gain a competitive edge rather than real value-added approaches. Anti-trust law creates uncertainty that can deter innovation (should we develop this technology if we can't fully profit off of it?)
Government institutions are like all other institutions - they want to survive. And to survive, they need to grow. Mission creep is inevitable. So even if you have identified a narrow area in which government intervention can help, it is not obvious that we should allow government to intervene.