There are a few things to realize about the US governmental system, that Europeans often don't get. The first is that these shutdowns are not that unusual. There have been shutdowns of some kind under pretty much every president in modern times. (Of course that begs the question "Why hasn't there been outcry every time one happens", but people do get used to things.)
Second, only the Federal government is affected, and the US have fewer services delivered by the Federal (i.e. national) government than other countries. A lot of day-to-day services are delivered at the state and municipal level. And for some essential services provision is mandatory, meaning the workers have to show up whether they are being paid or not.
Third US politics is adversarial. Really adversarial. A huge number of supporters of both political parties think that the other party is utterly evil and out to destroy the country. This means they will endure a lot of inconvenience if they think it's aimed at thwarting the plans of the 'enemy'. In most other democracies, especially European, most citizens would agree that the most important job of the government is to make sure that the country actually runs smoothly, and that ideological programs come second. (This isn't always true, but it's a lot more true than in the US.) The adversarial approach means that some party supporters see any form of compromise as 'siding with the enemy'. While in most democracies coming to a sensible compromise is a political win for both sides, in the US it can be a political loss for both sides, leading to more support for the extremist wing of your party.
Fourthly the US system if government is virtually guaranteed to have power split between its legislative bodies, and means that the three main parts of government - President, Senate and House of Representitives - are usually controlled by two different parties, and since the system give virtual veto power to each part, deadlock is inevitable. The US public has been constantly told that their system is "The best in the world", so they assume that these shutdowns are just inevitable.
Fifthly most Americans don't have any real exposure to political systems outside their own country, and what there is is often focussed on pointing out flaws (real or imaginary). This means they don't realize that shutdowns like this don't happen everywhere.