Actually you added the qualifier that jumped out at me when you started listing dictatorships with democracies. The "Free" election of individuals (although you did put free into the qualifier for the legislators).
And there is another slight error. The Westminster system is not a majority rule as neither of the 3 largest ones using that system at this time have a party that actually received a majority of votes, just a plurality.
But the popularity of the Australian government over the American one has to do with the biggest difference between the 2. In the Westminster system, the party winning the most seats rules. While we have elections for seats and then for a ruler. So they can (and often are) of different parties which leads to bickering and thus low esteem in the public's eyes (but gratitude to the people who beleive that the best government is the one that governs least).