I think you understate the problem when you say evangelicals "neglect" the matter that was crucial for the Jesus of the gospels, namely the care for the sick, the poor, and the downtrodden. You suggest as much in your next sentence. Indeed, the problem is that evangelicals are committed not to Jesus's ethics but to that economic feature of Americanism, to economic orthodoxy, that is, to free-market ideology despite its social Darwinian implications.
You speak of "thoughtful evangelicalism," but that strikes me as an oxymoron. I'm sure some evangelicals are highly intelligent, but I suspect they're confused about their religious options or are socially trapped in a simplistic theology. Why would a thoughtful Christian choose to be an evangelical when there are so many other Christian denominations? Indeed, Eastern Christianity in general is more thoughtful than the Western kind (just as Eastern religions are much more philosophically respectable than Western monotheistic ones).
Evangelicalism is fundamentalist Protestantism, a stripping down of intellectual complexities. This is a religion fit for the ignorant masses, not for intellectuals--unless the latter are actually committed to a form of Machiavellian or Straussian doubletalk. Often, the intellectuals in a social movement are the most cynical members, so they propagate noble lies to protect the faith of the more simpleminded folks, because the truth is mind-shattering and fit only for the intellectual elite (as Leo Strauss explained).
The gospels hint at precisely that esoteric-exoteric distinction, with Jesus's explanation of why he speaks in parables (Mark 4:10-12, which quotes from Isaiah 6).