Among the attributes you mention, while there is no reference to "diseaseless" and "brain", there is a popular verse in the Gita that refers to multiple lifetimes , attainment of wisdom and its rarity, in the context identical to that quote. It's possible Sri Paramahamsa Yogananda-ji re-paraphrased it to a more modern context and for effect of persuasion.
Bhagawad Gita, Chapter 7, Verse 19 :
bahunam janmanam ante jnanavan mam prapadyate vasudevah sarvam iti sa mahatma su-durlabhah
After many births and deaths, he who is established in wisdom
surrenders unto Me, knowing Me to be the cause of all causes and all
that is. Such a great soul is very rare.
While explaining the fate one who has failed in Yoga, Lord Krishna mentions continuity across births in spiritual effort and that he/she will take it up further from where they left, till they attain SamSiddhi (perfection )
The reference to the perfecting the human brain, may be towards the development of a keen and sharp intellect that , rather unrelentingly, differentiates between the Eternal and the Ephemeral, Nithyam and Anithyam, on a constant basis, referred to as Nithyanithya Viveka Vicharam in Bhaja Govindam, verse 30.
In the Part 2 of the book The Path of Sri Ramana, written by Sadhu Om, the author refers to the entire spectrum of evolution of a human mindset as akin to a school, spanning births. From the stage of animality (the lowest, i.e. kindergarden) , gradually to higher stages, such as bargain prayers, ritualistic worship, multi-god faith, one form and one name (Ekagrata Bhakthi), Darshan of that chosen deity, and eventually, even the withdrawal of the deity so that the focus may turn to the contemplation formless Brahman.
Finally, taking "brain" for its physical meaning, it has been known scientifically using electro encephalogram studies, (definitively, and not speculatively), that meditation and the dwelling on the inner consciousness changes brain activity for the better. To show that it is not a speculation, here is a reference to one of many research journal articles : (PDF)
Swamiji, could also have been referring to "man", in the sense of "Man", that is Mankind, as to the collective evolution of mankind towards a state of consciousness-aware life.