No, but the point is that anomalies always exist in any voting system, and it's a misunderstanding to think that "democracy" is tied to a particular system.
Perhaps so, but this is a long way from saying that every system is democratic or equally democratic.
It sounds like you don't really understand what "simulates a series of elimination rounds" means. Everyone gets to vote in every round. If your first preference is still in the running, you get to vote for that. What, you want to vote for something else as well? Or you want you vote to weigh more? If anything, it's the people whose first choice has been eliminated and who have to choose from the remaining candidates who should be complaining.
Your opinion of my understanding is, of course, your own, as are your implications that I consider my vote should be more important than the vote of others.

My point by saying that everyone's second vote should count (you could read that as just mine if you're determined I suppose!

) is this. Why not take the first and second vote of all voters and not eliminate anyone, rather than only permit those with whom the fewest agreed to change their vote? Stilll no winner and it's the 1st, 2nd and 3rd votes etc. This also means that all candidates stay in so no-one needs to complain that their candidate was eliminated. After all it is possible that everyone who did not vote for the bottom candidate as their first choice may have voted for them as second choice.
No one's vote (including mine 8)) counts for anything more or less than anyone else's.
Let's say we were trying to decide where to hold Mittens next year, and we had three proposals with different number of people in favor:
Canada - 12
Spain - 10
Netherlands - 5
Now, there's not a whole lot of support for the Netherlands, so let's rule that out, and we might get:
Canada - 12
Spain - 15
(Because the people who wanted to go to the Netherlands would rather stay in Europe.) Has this been unfair? Undemocratic? Have the people who originally wanted the Netherlands had "more votes" than anyone else?
They are the only ones who have been able to use their
second vote. I do not accept that, even after their country (candidate) has been eliminated, their first vote is worth anything less that any other vote for an unsuccessful country (candidate). They have therefore had two "goes" when others - including others who were not "successful" - have not.
Obviously the option is still open to still only vote for 1 country (candidate) if that is the only thing you want.
Is it fair or democratic that some people's second (&third etc.) vote counts and other's don't? In my opinion the answer is No.