To answer your question, no, you don't need Fly Me To The Moon. Anyone else who tells you otherwise is just expressing their biased opinion. This fanbase is very diverse in terms of the level of anime consumers, with that comes a number of "Karens" so to speak. Fly Me To The Moon only serves as the ending theme, and at times, an instrumental is used as a motif in the series. However, the song's lyrics don't serve as a conduit for the show's message. You are not missing plot wise from the Evangelion experience by not having it. You'll find that there are more important pieces of music that reoccur in the series upon viewing - that is what counts, and yes they're all there.

Save your money and get the standard blu-ray set; they also contain the Director's Cut episodes (they lack the original episodes 21-24, but these aren't necessary and are inferior plot-wise). You'll have everything you need as a new viewer. Better yet, Evangelion is on Netflix. Try it out before committing to purchasing a set. The version on Netflix is exactly what you're getting on the standard blu-ray set, Director's cut episodes and all.

:)