As you've suggested, there's a lot of improvisation in jazz. Almost everything is open to interpretation by the musicians--including melodies, rhythms, feels, and specific voicings for each chord. For most genres of jazz, the chords are usually the most fixed/immutable component of a song. (That said, chords are not completely unchanging. Chord substitutions and playing outside both introduce on-the-fly changes to the chord progression.)
In addition, there is a rich tradition in jazz of learning by listening to famous recordings. This old tradition extends far into modern-day jazz. As a simple example, imagine you're on a jazz gig with some musicians you've never played with before. The saxophonist calls the tune Autumn Leaves. One of the first things you would ask before the group starts playing is, "Do you want to play the Miles intro?" Everyone would know this intro because it's such an iconic version of the tune.
Turning our attention back to lead sheets: you're on a gig and there are ~300 or more songs you might be asked to play. Only some of these songs have famous intros, feels, rhythms, etc. that you're expected to know from famous jazz recordings. From a publisher's perspective, it wouldn't make a lot of sense to write out all of those introductions along with the chord changes. What if a song has two different famous interpretations? What if, over time, musicians change which interpretation they favor? There is likely just too much subjectivity in picking and choosing particular arrangements to include in a book with lead sheets/chord changes. And doing so would abandon the original song's core, which the sheet music is supposed to represent.
In many cases, pianists use lead sheets to jog their memory. They know the songs well and know how the recordings sound, and can use that as a guide. But they haven't memorized every single chord and thus use the lead sheet as a memory aid when they forget a chord or two. In other cases, a band might start playing a tune that the pianist has never heard of. The lead sheet provides the essential bones/structure from which the pianist can then improvise his/her part to match the dynamic of the group he/she is playing with.
So a jazz pianist likely isn't improvising everything when he/she looks at a leadsheet. Certain things are remembered from recordings, while other things are created spontaneously. This isn't as intimidating a feat as it sounds--one achieves this on-the-spot improvisation by drawing very heavily on the practice one has done. Taking rhythm as an example: one must improvise rhythms to use when playing chords behind the melody. But many of the candidate rhythms comes from a relatively small handful of rhythmic figures that sound good (e.g., dotted quarter followed by an eighth note). So practicing these known rhythmic figures makes the left hand automatic when supporting the right hand and its melody.
