this view of an average Americans level of agency in our political process could most politely be described as 'simplified'.

Before you even get in the weeds as to how voting actually works, that two party thing you mentioned provides, at the ideal best, two people with two sets of ideas about how to get things done. IF they don't agree on how to manage something (say we get the rare politician who's not in favor of further deregulating banks), there's no guarantee that one of those two ideas will be any good, or aligned with the political zeitgeist they're supposed to represent. Much more often, the wedge issues they differ on are a fraction of the things they'll be making decisions about, and stuff like military spending, bank deregulation, and general market deregulation are bipartisan supported. So if you don't like either of those, you're effectively excluded from the political process.

Then, you have all the ways in which the voting process is gamed by those in charge of it. Gerimandering is an obvious and easy one to spot, but there are other, more pernicious methods employed - like restricting voting to a single day, not making voting a holiday, purging voter rolls prior to the primaries (we'll deal with the primaries in a second), restricting felons from voting (whilst making things like marijuana possession a felony, an easy way to shunt out the underprivileged). There's more, for good reading on this topic check out Republic Lost by Lawrence Lessig.

We are married to first-past-the-post voting, which is demonstrably the worst system but the easiest to game.

The supreme court, which makes many of the most impactful political decisions, isn't voted in. And they have life terms. So even if you manage to make progress elsewhere, they can just decide that things should be different and change the way the law works.

The Senate arbitrarily distributes the power to block legislation among the least populous states, making it easier to manipulate the population into voting for people who will do this.

K Street (the lobbyists) literally get paid to 'help' draft legislation that favors only those who can afford it. Some of our laws are actually for sale.

Then you have the primaries, which are effectively controlled by two fairly small cabals of (who exactly? I dare you to enumerate the people involved). Each of which has it's own set of obtuse rules that's not subject to an iota of the scrutiny that actual elections are subject to, even though you could make the argument that the primaries are the more important election, because they decide what the two options will be.

Then there's the news. Rather, there's the institutions that claim to be news, but all of which behave more like winged propaganda machines, pumping their color and fueling anger at (whatever the wedge issues are for the other team today). These outlets all have things they will not talk about, like the issues with our elections or growing income inequality, because these things are precisely the things that the power structure, which all major news networks benefit from, is founded upon.

One thing both parties agree on completely; there should only be two parties. This would strike one as odd if they weren't so deeply acclimated to the situation.