It's good to hear that I am not alone in some of my wacky views. For far too many years, I've had a project on the influence of Nietzsche on the positivists in the back of my mind, but my feeling that I'd need to work on my German and that I had too much else that was of higher priority held me back. With the dissertation about done, I need new projects, though, and I also feel that it is long past time I did work on my German. Anyway, scholarly investigation of the Logical Positivists is getting trendy; I wish I'd gotten in on the ground floor of the trend, but I'd better exploit it before it goes away again. I suppose another obstacle is that, as Jim noted, some of Nietzsche's influence on the Logical Positivists was surely indirect, via Wittgenstein, and my enthusiasm for serious investigation of Wittgenstein is not so great.
Jim's comment also cited Carnap's paper under the standard English translation, as "The Elimination of Metaphysics through Logical Analysis of Language." This is in no way misleading regarding Carnap's purpose, but "Ueberwindung," the first word of Carnap's original German title, was one of Nietzsche's favorite words, and I don't know if it's ever rendered as "elimination" in any Nietzsche translation; Kaufmann usually (always?) uses "overcoming." That's why I suggested that the title was somewhat Nietzschean, a fact obscured in the standard translation. Issues like that are among the reasons I really do need to examine the sources in the original German for my project.
In any event, the recent scholarly work has noted that pre-Nazi German philosophy involved considerable mutual influence among the competing schools. The sharp analytic/continental divide was greatly helped by the way the leading analytic figures on the continent almost all fled the rise of the Nazis, producing a geographical divide between the schools which served to help heighten their distance in less literal respects. Thus, it is probably in general misleading to impose the analytic/continental distinction on pre-Nazi philosophers, and if I'm right about Nietzsche being a major influence on the Logical Positivists, that would be further evidence of the untenability of the division for philosophers prior to the mid-20th century, since Nietzsche is standardly associated with the continental school.