But what happens when people won't compromise? You end up with the status quo ad infinitum, in particular around any especially contentious issue; abortion, gun control etc.
That's the claim, it is not necessarily the reality. It's interesting that you should mention abortion, because progress is finally being made on that issue in Ireland in part because we have a coalition government. The main party of the current government is socially conservative, they'd be happy to sit on their hands and not to legislate for it at all, which was their position prior to the last election. But they're in a coalition with Labour, a party who do want to legislate for it, so there's finally going to be legislation for it.
And if you want a British example, their coalition government is the reason why gay marriage is being legalised there. The Conservatives on their own weren't going to push for it, the majority of their MPs voted against it in parliament. But the Lib Dems wanted it, and Cameron needs their support to remain in power, so he went against the will of his own party and pushed for it.
So, I once again refute your claim that multi-party governments can't get things done on controversial issues.