In most democracies, the current coup attempt, would not, and could not happen. Parliamentary systems do not allow this. When there is deadlock, it does not lead to the government closing, but being dissolved. That sounds scary, but this does not mean what most Americans would think it means.
When our government shuts down due to deadlock, it doesn't mean our elected and appointed officials disappear, but many government services do. On the other hand, when government is dissolved in a parliamentary system, government agency continue to function as if nothing happened. It is the elected officials that may find themselves out of work.
When deadlock occurs, and cannot be resolved, a vote of confidence is called in the parliament, which will be most likely result in a vote of no confidence. The government - in the parliamentary sense, meaning the Prime Minister and his cabinet - is dissolved, and new elections are held quite soon. A caretaker government is appointed with one mandate - keep things running, but no new initiatives - in other words, the parliamentary version of a continuing resolution.
Usually, the new elections are held within a couple of months, and most other democracies have either total public financing of elections, or strict financing rules. These election are fast, much cheaper than American elections, and perhaps a bit more tastefully done.
The beauty of being able to change governments so quickly is twofold. 1. A administration that cannot govern doesn't get to hang around very long. 2. Because the elections are held so soon after whatever caused the deadlock, the party that is perceived as having caused the damage will be punished, as voters are not likely to forget these transgressions in a mere few months.
Our system doesn't make room for anything like this. It would require either a new Constitution, or a reworked Constitution, and I think that would scare too many people across the political spectrum to even talk about that, so I think we are stuck with what we have. But I still look at these countries and say to myself that there has to be some way to put a stop to these shutdowns.