I am from England and agree with you. Maybe it isn't as commonly used as "I'm fine/good/well" etc. but it can be used. Equally, one could replace "I am" with "it is" or "things are" depending on context. I feel like it should be accepted. Sometimes it is easier to learn the literal translation as it makes remembering French expressions easier.
"Je suis bien" : the meaning can be "I am a good man/woman"...if you use it as "I am fine" you put some other words after, and it is more a satisfaction for a special time : je suis bien dans mon bain (in my bath), dans mon lit (in my bed) or right now. It is not a generality.