In a system such as the US’, with no ranked-choice voting (the same lack of which is why the French in 2002 got stuck with Jacques Chirac as
Prime Minister president when they, instead of voting for Lionel Jospin, frittered their votes away on several fringe left parties and were then forced in the runoff to vote for Chirac to avoid the Nazi LePen), third parties can almost never be anything but spoilers. This is shown in how the Republican Party treats third parties on the right (which threaten to take votes from the GOP) and the left (which threaten to take votes from the Democrats).
Republicans love actual third parties, but only those that are likely to take votes from Democrats. This is why over the past couple of decades, Republicans and their big-money patrons have made a point of buying ads for various Green Party candidates (like Nader in 2000) — and have often provided (most notoriously in the 2006 Pennsylvania Senate race) the bulk of a Green Party candidate’s funding.
But when a third party or even a mere third party movement is formed that threatens to take votes from the Republicans, the GOP has a much different response. When the original TEA (“Taxed Enough Already”) Party movement formed in the late 2000s, it was formed not as a branch of the Republican Party, but as an actual, separate third party movement with secular-libertarian leanings. The Republicans could not allow this to stand, so they used an old trick — create well-funded “vacuum cleaner” groups to suck up potential TEA Partiers and keep them within the GOP fold before they could find and join the real TEA Party.
Republican activists, armed with mountains of cash, much of it from the Kochs via their backing of Dick Armey’s Citizens for a Sound Economy, which merged with another conservative activist group Empower America to become FreedomWorks, created a Potemkin movement in 2009 called “the Tea Party Express” to serve as a vacuum cleaner to starve the true TEA Party of members. Armey’s movement had lots of cash available for “Tea Party” candidates — but only if they promised to a) uphold the Republican Party platform, and b) caucus, once elected, with “like-minded representatives” (in other words, Republicans).
Thus was the TEA Party neutered and reabsorbed back into the GOP.