The key here is "who is supported by the dominant political party". The tea party gained so much clout because they got organized to contest internal elections

The key here is "who is supported by the dominant political party". The tea party gained so much clout because they got organized to contest internal elections and primaries inside the GOP. It's much more efficient to leverage the existing party infrastructure than to run as a solo independent or try to create a 3rd party. Party elections (for instance, county committee elections) draw much less turnout and can be swayed much more easily by an organized group.