Clay Shirky, in a Medium post titled "There’s No Such Thing As A Protest Vote":
People who believe in protest votes do so because they confuse sending a message with receiving one. You can send any message you like: “I think Jill Stein should be President” or “I think David Duke should be President” or “I think Park Eunsol should be President.”
Similarly, you can send any message you like by not voting. You can say you are sitting out the election because both parties are neo-liberal or because an election without Lyndon LaRouche is a sham or because 9/11 was an inside job. The story you tell yourself about your political commitments are yours to construct.
But it doesn’t matter what message you think you are sending, because no one will receive it. No one is listening. The system is set up so that every choice other than ‘R’ or ‘D’ boils down to “I defer to the judgement of my fellow citizens.” It’s easy to argue that our system shouldn’t work like that. It’s impossible to argue it doesn’t work like that.
Later in the same piece, he addresses complaints about the two-party system:
In America, by contrast, the coalitions are the parties. Our system also produces alternation of power, and requires compromises among competing interests, but those compromises happen within long-standing caucuses; issues come and go, but the two parties remain. This forces the citizens themselves to get involved in the disappointing tradeoffs, rather than learning about them after the fact. No one gets what they want in a democracy; two-party systems simply rub voters’ noses in that fact.
That last bit is super-important. The response I generally get when talking about the fruitlessness of third-party politics is that it would all be fine if only we had some form of proportional balloting, or if we got rid of the Electoral College, or had a parliamentary system. Leaving aside the obvious fact that such "solutions" are unlikely to the point of fantasy, they don't actually address the underlying issue.
You live in a country of 310 million people. Many of them disagree with you, some more strongly than others. To govern the place, you need to compromise with a sizable chunk of those people. The fact that the political system forces you to make those compromises does not mean the political system is broken. That's what politics is.