They do. However, I'm sure you can imagine an elected government acting in a way that the majority disagrees with. We're about to see it in the US (actually, we have for years if not decades). This is not just a US phenomenon, there's actual research showing that in liberal democracies, there's very little correlation between what the general public wants, and the policies instated by their elected officials. There is a strong correlation with the interests of the owning class though.

Here's a study for American politics: https://www.cambridge.org/core/journals/perspectives-on-politics/article/testing-theories-of-american-politics-elites-interest-groups-and-average-citizens/62327F513959D0A304D4893B382B992B

There's a Danish study as well. I'm having terrible trouble finding it though. It's an important addition because the democrats not representing the interests of the working class could in theory be a consequence of the US's two party system. The same result holding in multi-party Denmark shows that this is not the case.

At any rate, the point is that just because these countries are liberal democracies doesn't mean their population wants a US military presence.