The problem with "independent" voters (not those who have strong views one way or the other, but those who don't have any either way) is that they're fickle and deliberately choose not to educate themselves on political matters. A Democrat gets into the White House, they are happy for a bit then get tired of him, then vote for a Republican, and the cycle repeats itself. Because they don't have strong views either to the Right or the Left, they don't vote consistently for either Right or Left, but for whatever isn't in the White House, depending on if it takes them four or eight years to get fed up enough with someone (and the ideology, conservative or liberal, that guy represents).
During the year, they don't pay enough attention to politics, I suspect. A little bit of the local news once in a while, or a few articles in the paper - that's it. As both serious Right- and Left-wing voters know, it takes more than dipping your toes in the water to be swimming, and it takes more than the occasional cursory glance at news sources to be informed. So, to "reach out" to independent voters, what needs to be done?
Either abandon your principles and appeal to them on a contrary basis vis-a-vis the guy who's in the White House, or stand firm on your principles and trust that an honest display of them will touch the hearts and minds of independents, who also want what's best for their country, even though they don't see the need to stay informed as a daily practice.
So, the Tea Party is hardly an instant death-blow to a candidate in the faces of independents. All the independents need to do is see what the Tea Party and any associated candidate has to offer and where they stand. If they aren't Left-leaning independents, they'll not be put off and probably attracted. Since most independents, despite their general lack of political interests, lean essentially either conservative or liberal, that is what determines much of how they vote, I think.
Besides, what's so "extreme" and "off-putting" about the Tea Party, anyway? They only want constitutional rights and limited, responsible government. Oh yeah, sooo hardcore and sooo extreme

Only people who insist on letting the mainstream media influence their views could think that, not those who look at the Tea Partiers themselves and see just where they stand and what they think.