But there's never really been class warfare the poor against the rich. There has never been more than a trivial effort to eliminate the rich. We just don't roll that way. It's always been about letting labor have a piece of the pie. Not about taking the whole pie away.
Debatable. The US had an extremely vibrant and militant labour movement up until the interwar period, which naturally produced socialist currents, some of which, like the Debsian socialists or the IWW, achieved no small significance. The major difference between the US and European labour movements isn't that the former was intrinsically non-socialist, it's that American capital was able to supress or exile the socialist currents that did exist before the post-war compromise was established, with the result that they were more or less stricken from the official histories of the labour movement. You got card-carrying communist trade union organisers in both the US and UK in 1920- the difference is simply that only in the latter were they still there in 1980.
Anyway, like Cheezy says, class struggle ("class warfare" is largely rhetorical, in both approving and disapproving uses) doesn't manifest exclusively in revolutionary socialism. It's equally as present in, say, taking an extra five minutes on your break. Until you're actually at the "brb, storming the Winter Palace" stage, it's fundamentally just a question of scale. Certainly, it couldn't demand a concerted effort to eliminate an entire class, or the capitalists would be unable to wage it anything other than a suicidal fashion.
(And I'm sorry for these delayed posts, but my internet access is spotty right now.)