Learsfool, your description is excellent; no need for qualifiers. I would contribute to it only by approaching it from a somewhat different angle.
There is no possible way to exaggerate the level of skill possessed by an excellent jazz musician. Their understanding of, and ability to move around within the world of harmony, is simply astounding. As you point out, there exists a fairly well established "repertory" of tunes that working jazz players tap for their performances. This list of "standards" grows over the years with more modern tunes by contemporary/current composers. Learning the melodies of tunes seems like child's-play when you consider what it takes to improvise (spontaneously compose) a credible solo (new composition) within the unique harmonic framework of any given tune.
Consider a typical "standard" jazz tune with a 32 measure, AABA form such as "Cherokee"; a tune that is 32 measures long, with an 8 measure verse (A) which is then repeated (A), followed by another 8 measure section (B) referred to as the "bridge" which harmonically bridges the two A sections to the returning and last (A). This bridge is usually recognized by a seemingly harmonically abrupt and even unrelated harmonic center; but, usually sets-up the return to the harmonic center of the "A" section. Now, consider that each of those 8 measure sections is made up of a sequence of harmonic "changes" (chords). Each of those eight measures will have one, two, three or even more chord changes within that measure. The choices of harmonic changes (chords) that a composer chooses create a "harmonic rhythm" which gives a tune harmonic pulse or momentum; a sense of forward movement. Each of these chords, while related, has it's own "palette" of color choices (note choices) that it offers the improviser as he travels through this sequence of palettes in real time. The improviser then has to choose from each individual palette (chord) and play notes that not only are found in that particular palette, but also relate to the palette before it, the palette that follows it, and most importantly the entire "box" of palettes (AABA/32 measures). The best improvisers can choose colors (notes) that are not found in any given palette (they are outside the harmonic center of that particular harmony) and still make them relate to the overall harmonic scheme. Consider just how little time an improviser has to react to the constantly changing harmony, and still choose notes that are more than simply "correct" notes and actually add musical meaning to the improviser's spontaneous composition.
Musical meaning; THAT is what seperates the "good" improvisers from the greats and what defines a true artist. It's possible to master the mechanics (physical and intellectual) of the process of improvisation to an extent that can be very impressive, but still shortchange the listener with an absence of music. While this can be subjective, the educated listener usually knows it when he hears it (or doesn't). This is the main reason that a more in-depth understanding of the music making process helps to appreciate who the real artists are. A visceral, non-analytic, appreciation of music is great and truly the bottom line; but, understanding the proces better can clear the way for a deeper understanding of the art.