When people disparage third parties as irrelevant, they're absolutely correct, except in the well-known case of a spoiler in the election. Over the entire country's history, there have only been a handful of congresses during which they held any direct political influence at the federal level (outside of the spoiler effect). Since the civil war, any direct influence has essentially disappeared.
A couple things to take away from these graphs:
1) Independents have been essentially extinct in the federal legislature since the 75th congress (1937-39)
2) The periods when they did hold any number of seats were when one party was very dominant; there has never (since the civil war anyway) been any real need for the ruling party to form anything resembling a coalition with them
The reasons for this are numerous, and have almost nothing to do with the quality of the two parties or the quality of the third party options, and almost everything to do with how the elections are structured. It's not one of those things where if everyone just believed in magic and stopped voting strategically we could have a multiparty legislature; that could only really happen during the midst of one party's catastrophic collapse, and only very briefly. This is a good UN-affiliated site to look at different election systems and what effects they have on elections, with historical examples:
aceproject.org/ace-en/topics/es/defaultIn essence, it's the FPTP system that ends up "wasting" votes that aren't cast for whoever gets a plurality in their district, an effect magnified by us having extremely large districts (the 2010 apportionment gave an average of 700,000 people per district, compared to around 30,000 in the early 20th century). If all the votes were tallied at the state or national level instead and given seats based on the actual proportion of votes (i.e. "proportional representation"), the two-party system would die almost overnight (and we'd stop having irregularities where the party with fewer national votes ends up with more seats due to things like gerrymandering and the apportionment process between the states).
And I say all this as someone who usually votes third party in federal elections; it is throwing your vote away, but the real joke is on the people who vote for one of the other parties without realizing that they're also throwing their vote away. Only a small fraction of congressional districts and senate elections are actually competitive, and your vote dies in whatever district you're stuck in. Voting against whoever's locally dominant is just as pointless as voting third party, and likewise tossing your vote on the winning avalanche doesn't make any difference either. Now, if you live in a district or state where the election is not decided in the primaries and nobody knows how it'll turn out, then by all means you should vote for one of the two parties. But if like most people in the country you live in some Lordship's hereditary incumbency that will only be interrupted by death, redistricting, or the intervention of God, you really may as well be voting Mickey Mouse.
Or at least at the federal level, local and state elections are usually less hopeless (but only really because they have smaller district sizes, you're still stuck with whatever is put on your plate).
Edit: And it should also go without saying, the system that determines which parties can win in the most powerful body (the national legislature) will also determine who can win in the similarly structured state elections, and ultimately other offices like the president. It's only at the local level that the electorate is small enough that they can deviate from the national norm and have interests/positions/ideologies that are politically viable there but not at the national level.