My country of residence, Luxembourg, has a parliamentary system, which works similarly to others as described above and has similar advantages.
To illustrate using our last national election, from 2023, as an example: The election was held 8 October. The leading party, the CSV (centrist, right-leaning), did not win an outright majority and could not declare a government, so negotiations began to form a coalition. The newly elected members of parliament (which here is called the Chamber of Deputies) were sworn in on 24 October, but beyond an agreement in principle that the new government* would be led by a partnership between the CSV and the second-largest party, the DP (centrist, left-leaning), the coalition discussions were not yet concluded and the previous government continued on a caretaker basis. An agreement was finally reached, and a coalition Cabinet formed, on 15 November (this is unusually long by our standards). The official announcement happened the next day and the new government took over a couple of days later. No particular swearing-in was required; the coalition was created as a matter of agreement between the two parties who would govern jointly. (The Prime Minister does present his Cabinet to the Grand Duke as ceremonial head of state, but it’s purely a formal rubber stamp in our constitutional monarchy.) And the vast majority of the administrative apparatus under the ministerial level (beyond a handful of deputies who came in with the Cabinet) is staffed by “public sector” employees; the terminology is a little complicated but these are essentially service professionals whose tenure is independent of election cycles and who are not appointed on a partisan basis.
*As a note on definitions, I am using “government” in its European parliamentary sense, not in the American sense. For more information, see here. Basically, as mentioned by others, the American model, in which the “Presidential administration” comes in with thousands of appointments to supervise and implement policy, doesn’t apply here. The government sets policy, and the public sector is expected to implement it.
One additional advantage of the Luxembourgish system, not really touched on by anyone above, is that these coalitions are commonly reflected in the makeup of the Cabinet. Other countries with multi-party parliamentary systems are frequently led by governments formed by party coalitions, so that’s not unique here. But the semi-proportional mechanism by which we elect our parliament means it’s very rare for any party to achieve an outright majority, and the government is almost always a coalition. (Compare the UK and how its seat-by-seat FPTP model produced a Labour majority and a purely Labour government.) If you followed my links above, you will have observed that the current Cabinet membership is almost equally balanced between two parties — eight from the CSV, seven from the DP.
This has a naturally tempering effect on the aggressiveness of policy initiatives, as you’d expect. Without an absolute majority, compromise and negotiation is the name of the game. Again, this kind of arrangement does commonly happen in other countries, but in Luxembourg, it’s baked into the system almost as a given. Our political culture, as a consequence, is a mixture of conservatism and progressivism: we tend to move toward change quite slowly, and we do, often, have trouble with nimble adaptation to rapidly changing circumstances; but we’re almost always moving forward, and never, ever moving backward.
Politics here is pretty boring, without the volatility you see in the US. And that’s the way it should be.