"And again: who decided your PM would be PM? Voters? Or was it an internal decision made by a party without voter input? How can you claim that the US is less democratic than that?"
It's not a perfect system, either, but the PM has to be among those elected to office - it's entirely possible (though rare) for a party to win the election, and their candidate for PM to lose their seat (and thus be ineligible).
It does also offer a semblance of "honesty" or vested interest. There's an accountability - if you as the PM are not capable of running a platform or country that makes other members of your party electable, they can and will remove you. So, at least in theory, there's a feeling of representative power. Most party-based removals from positions have come directly from the people, unhappy with performance, and then groups of politicians saying "Yeah, we're out of a job if we follow you/keep down this road".
It does also have Survivor-like elements, for better, and worse.
But the candidate for PM is known by the public, so in a sense it's de facto voted for.
Just like in the US, to rephrase your last: "Who decided your President would be President? Voters? Or was it an internal decision by an electoral college who in some cases is not obligated to respect the wishes of those they represent? How can you claim that to be democratic?"
As someone who has lived substantial periods in both Australia and the US, I have a decent perspective.