Notwithstanding Article 5.2 on accountability, I do not believe that non-controversial data processing under legitimate interest needs to be well documented in practice, although I may be proven wrong. I believe it is sufficient to mention legitimate interest and what your legitimate interest is in broad terms in the privacy notice.
For controversial use, that is processing which the data subjects may not approve you of doing, I believe you need documentation on the balancing test. However, in controversial cases you probably also need documentation on the necessity both when it comes to performance of contract and legitimate interest.