The word "mindfulness" means "recollection"; to "remember" or "bear in mind". MN 117 says:
One is mindful to abandon wrong view & to enter & remain in right
view: This is one's right mindfulness.
So yes, the aim of mindfulness is to keep the mind free from craving; thus to keep the mind in pure consciousness. Pure consciousness means a consciousness without unwholesome defilement.
When this is done, yes, attention is never directed onto anything (apart from to keeping the mind free from craving). Attempting to direct attention onto something, such as the breathing, is probably the greatest obstacle to stream-entry and properly practising Satipatthana.
When there is no directing of attention, if consciousness is pure, it will automatically connect with the breathing (because the breathing is the grossest sense object for a pure consciousness). Although the suttas describe observantness of breathing, feelings, mental states & Truth, the sutta do not teach to direct attention (except to having no craving).
MN 118 & SN 48.10 say, for example, the development of mindfulness, concentration & jhana relies on letting go ('vossagga') rather than the directing of attention. Mindfulness remembers to abandon craving; to let go. When this letting go of craving occurs, consciousness will be relatively pure and can automatically progress to greater purity.
Ajahn Buddhadasa explained:
As for samadhi, an empty mind is the supreme samadhi, the supremely focused firmness of mind. The straining and striving sort of samadhi isn't the real thing and the samadhi which aims at anything other than non-clinging to the five khandas is micchasamadhi (wrong or perverted samadhi). You should be aware that there is both micchasamadhi and sammasamadhi (right or correct samadhi). Only the mind that is empty of grasping at and clinging to 'I' and 'mine' can have the true and perfect stability of sammasamadhi. One who has an empty mind has correct samadhi.