Yay, yet another first-person-pronoun question!
I know that 自分 is commonly used as a personal pronoun in indirect speech, e.g.:
マイクは、自分はなんと不注意なのだろうと言った。→ Mike said that he was very careless.
スミスさんは自分はニューヨークの生まれだと言った。→ Mr Smith said, "I was born in New York."
as well as a reflexive pronoun (for speaker or listener/third party):
自分を信じて！→ Trust yourself!
and as possessive, to emphasise the possession:
掘っ建て小屋でもいいから自分の家が欲しい。→ I want my own house, even if it's a shack.
All these make perfect sense and share the same underlying meaning of myself/yourself/oneself (with emphasis on the self). But I have also encountered (in fact even caught myself saying) 自分 in places where 私/僕/あたし/etc could have been used, with only a very weak emphasis on "myself" (e.g. not answering a "did he or did you?" question):
自分はスポーツが嫌い。→ I hate sports. (?)
My question is, is this use as a general gender-neutral 私 substitute correct (in cases where there is no particular need to assert oneself against others)? And if so, what would be its speech level (e.g. could it be used in a formal context)?