Just on this partDepending on how you've learned to read music, it is possible to polish and interpret a piece without ever memorizing it. The idea that piano music must be memorized became popular and then the norm at a certain historical point, and there are some who dispute it.
However, it is one of the most effective means to force one to use the aural image in the mind to be the primary coordinator for a performance rather than a secondary one.
Even when reading from a score to perform, the aural image should be the primary guide for an effective performance; however, many struggling beginners taught classically use the the visual score as the primary coordinator.
Memorization is one of the means needed to break that habit.
Also when polishing from a score, it is absolutely critical to work on areas where the score is most likely to distort due to the visual presentation.
Roughly: barlines, beaming groups, ends of staff systems on a page, page turns, splitting of registers across the hands/grand staff, etc...
It's a minefield! And strategic memorization of those specific areas is one of the best ways to combat the tyranny of the barline and its compatriots.