The other question is: How democratic are those parties? - he primaries seem as such ...
I honestly and genuinely don't think that's much of a question.
I encourage you to read Lawrence Lessig's "Republic, Lost"[1] - which talks about how the big money behind campaign financing makes average Joe's feel as if they're choosing a politician based on their will, which turns out to not really be the case. Rather, they're choosing from a select group of politicians already rubber stamped "OK" by deep pocketed donors.
A good quote (not verbatim) from the book is basically how deep pocket donors have a sense of: "Let me choose the potential candidate options, and I don't really care who wins".
So it's the "will of the people" only so far as the people's will aligns with something they have 0 control over - the intentions/ambitions of those with money - which doesn't sound much like free will.
[1] https://www.amazon.com/Republic-Lost-Version-Lawrence-Lessig...
But I'd say the 2016 election clearly refutes your assertion.
Republican fundraising as of 6/22/2016 in USD$millions, {total}, {candidate}, {affiliated PAC} [1]: Donald Trump (67.1, 64.6, 2.5), Jeb Bush (162.1, 35.2, 126.9), Ted Cruz (158.0, 92.6, 65.4), Marco Rubio (125.0, 47.3, 61.8)
If one wants to blame the Illuminati for secretly supporting candidates... this is a harder point in history to find support.
[1] https://www.nytimes.com/interactive/2016/us/elections/electi...
Now, if the US switched to a radically different voting system today, we probably wouldn't see a flourishing of outside parties tomorrow; it could take a little while. But people sympathetic to outside parties such as the Greens or the Libertarians would be far less reluctant to vote for them if it didn't mean splitting their vote and causing the "lesser of two evils" to lose their local district.
Preface: I'm happy to be corrected, since i'm not very well-versed in this topic.
In the US, is it even a democracy when a new party can't come up and win an election? You can legally lobby for a party which in reality is just a fancy word for a bribe, so how do you expect either of these parties to have a common's person best interest in mind when it comes to making policies?
Compare this to India, where a person went on a strike, and then a group of people formed a party with the said person and he ended up becoming the Chief Minister of Delhi. Now, did he turn out to be good or not is another debate, but my point is, people were fed up with established parties, and it was actually possible for a new person to form a party and make a government out of it. Majority of the population wanted a change, and it was possible for them to get it.