In most democracies that don’t suffer from the two-party system, there is as far as I’m aware nothing like the American primary/caucus system. The parties just decide who their candidates are with no input from anyone but dues-paying party members.
The last UK leadership election for the Labour Party [0] had a turnout of 500k+, or roughly 1% of the population. I won't bore you with the details of whether it's correct to think of all 500k as "dues-paying members".
That's a lot less than the 30 million or 10% of population [1] who voted in the last Democratic primaries in 2016, but not "nothing like" it. And we could pick other years the turnout was much lower, e.g. 7 million [2] in 2012 where Obama did not have meaningful opposition).
UK is not a good counterexample here. It also uses the first-past the post voting system, which trends towards two parties since any third party becomes a spoiler.
With a different voting system there would be a wider range of smaller parties, so you don't need to wrest control of a major party to be represented.
