Is ~を好く used and is it even grammatical?
It's grammatical, but hardly ever used in the form 好く in standard modern Japanese.
However, the passive form 好かれる is quite common in standard Japanese.
Dialectal negative forms 好かん or 好かへん are also quite common in certain parts.
If it is not used, why is it so?
This is mostly me guessing, but I imagine that people shied away from the use with the direct object marker を because it seemed a bit too... well... direct.
Why ～が好き instead of ～を好く? I think this might be for a similar reason that ～が嫌い exists next to ～を嫌う. However, maybe it's more acceptable to be direct about things that you dislike than things that you like, so ～を嫌う survived.
But then why not ～が好く with が on the object like many other verbs that have to do with emotion? (Incidentally, ～が好かん takes が so this would fit into the pattern.)