The push to remove the 14th Amendment flies in the face of what this nation was founded on. "E Pluribus Enum" means "From many, one" and is on our currency for a reason. We are all, in one way or another - immigrants. I'm not sure when "Give me your tired, your poor..." turned into a "No Vacancy" sign, but it seems to have done so.
But there is a bit of slight at hand going on here. While the 1st section is the one that everyone is getting up in arms about, I don't think that's the end game. It's hyperbole, used to elicit an emotional response and push the logic to the back of your head while you yell that someone's taking a job you honestly weren't be willing to do in the first place.
Think about it for just a moment: Why would people who don't trust the US government for anything, suddenly want to give that same federal government the ability to decide who are and who are not citizens? What would the metrics be for that citizenship? And how could we prevent political parties from using it to shape the political landscape? It exists for a reason, because it was abused in the past. Slaves who were freed were not allowed to be citizens because they weren't born here.
And the really REALLY tough question to answer honestly: Do you think that I, a white male - would EVER have my citizenship status questioned by anyone if the 14th Amendment was overturned? We may not want to admit it, because it's disheartening; but we all know that answer.
That's not the real purpose behind this provision. The real issues at hand are Section 2, which is about state representation in the federal government and the electoral college.
Section 2 states that "Representatives shall be apportioned among the several States according to their respective numbers, counting the whole number of persons in each State, excluding Indians not taxed." Now if you're a Native American, you're probably not too happy about the verbage; however, if you're a career politician in a solid blue or red state (especially a solid red state, as they tend to have fewer representatives, and therefore fewer electoral votes) then you want this gone.
Take CA and MT for example. CA is a solid blue state, with 55 electoral votes. By comparison MT has 4 electoral votes. Why would getting rid of section two be helpful? Then states could petition for a new system of electoral voting (or more representation in the house) and negate the power of the more populated states (who also tend to vote Democratic) When you tally up the 'solid' states electoral college votes, it's almost impossible for the Republican party to take a presidential election. They've lost the war of 'voting with your feet'. There's only 12-13 "swing states" that decide an election. If you consider the number of states that have voted the same way since 1992, the Democrats 'start' with 243 electoral votes to the Republicans 102.
With only 13 or 14 swing states, that means with the 14th Amendment intact, the GOP would have to win ALL of them, whereas the Dems only have to win 1 or 2 states in order to secure a national election.
But if they told voters that, it'd have no traction. But scare people silly with talk of illegal immigrants hopping borders - they don't think twice. Repeal it, and damn the unintended consequences!