A relevant story
I was in a lesson once with Arturo O'Farrill, a brilliant jazz pianist. He asked me to play a ballad, and when I was finished, he said something like "for that song, you can take it further by doing this sort of thing." He proceeded to play the same song using ridiculously cool chord substitutions. My mind was blown, and I assumed he came up with all of this on the fly.
When I got home that day, I looked up his albums and found a solo arrangement of that same tune. The recording was a few years old, but when I listened to it, I heard him play the exact same crazy voicings and reharms from our lesson.
Arturo wasn't being inauthentic in any way at all--he never claimed to create the arrangement on the fly. I just had been carrying the same misconception you possess: that everything in jazz is spontaneous.
The jazz piano "greats" all planned out their solo arrangements
With solo piano, it's extremely common among highly talented jazz pianists to work out full arrangements in detail. This entails planning out which voicings to use, which reharmonizations to deploy, the moment when a walking bassline should begin, where to place rootless chords, etc. If you need to write out the arrangement to remember it all, then this is fine. However, that's probably an indication that you need to spend more time practicing those piano voicings in all 12 keys.
As you work through more and more songs in this fashion (planning an arrangement for the entire melody), your solo piano chops will improve, and your playing will become more fluid. You'll find that, when you arrive at a dominant 7th chord with the 13th in the melody, you have a few different tricks up your sleeve that you could deploy.
This applies mostly to the chords and arrangement of the melody. Even in solo jazz piano, it's not very common to plan out full improvisations.
Playing in a group is a totally different beast
In general, practicing your solo chops won't be much help for playing in a group. If you walk into a small combo and start playing Bud Powell voicings (which contain the root) in a lower octave, the bassist will give you a funny look. If you start walking a bassline with your left hand, you'll get more than just looks. Moreover, when playing in a group, it's far less common for a jazz pianist to enter a song with a plan for every voicing.
There are good reasons for these differences: in a solo context, the piano is far more constrained because it must fulfill the role of bass, harmony, melody, and beat--all in one. That's why advanced planning is so needed. But in a group context, the pianist only fills the role of harmony (because the bassist provides the bass, the horn provides the melody, and the drummer provides the beat). That frees up pianist to be more creative and responsive. And in general, there is a greater expectation in a group setting that the pianist will listen to the rest of the band and respond/adapt accordingly.