In most other democracies, the shutdown/coup attempt/hostage taking that is now thankfully over could not happen. Most first world democracies are parliamentary systems, and have safeguards that prevent this happening. 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 agencies 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.
Since my original post, I have thought of a possible solution to this mess without a politically impossible major revamp to our constitution. Here is my first solution. It is not perfect and it will not prevent all shutdowns.
Most of these shutdowns happen, because Congress wants a budget that the executive branch does not want. What if we had a constitutional amendment that requires that if there is a deadlock between the houses of Congress, or between Congress and the President, there will be an automatic indefinite continuing resolution until a new budget is passed. Congress could only defeat this with a super majority of 2/3 in both houses. No shutdown could happen without the overwhelming support of Congress, which probably won't happen unless the President loses all credibility with Congress, or with the opposing party having 2/3 of both houses. Either way, if Congress can pull this off, they will own the problem and have to solve it. If Congress cannot override the CR, then it will stay in effect until a real budget is passed. What if politicians of either party want to accept this indefinite CR as the new normal? That would not work, because it would be impossible to pass new laws that required funds to be budgeted. Likewise, discretionary spending could not be adjusted for inflation. And operating on a CR long term could be come a campaign issue against the party perceived responsible for the situation. Maybe my idea is totally naive and unworkable, but it's this idealist part of me that believes that there must be a better way than this.
My second solution which is hopefully taking place as I write is this. The Republicans will hopefully pay such a dear price for this mess, that it will be at least a very long time before this happens again.