In a way yes, and in a way no.
All of mankind can properly be called children of God, insofar as He is the active cause of their existence and are made in His image and likeness.
All those who have been born again are properly said to be children of God, in a deeper, spiritual sense. Through this relationship they call God Father and Christ Brother.
After the fall man resembled Adam no longer God
It may make you feel good to believe that God looks on all men as His children, but it just is not so. See how Jesus answered that very point
Jhn 8:40 But now ye seek to kill me, a man that hath told you the truth, which I have heard of God: this did not Abraham.
Jhn 8:41 Ye do the deeds of your father. Then said they to him, We be not born of fornication; we have one Father, [even] God.
Jhn 8:42 Jesus said unto them, If God were your Father, ye would love me: for I proceeded forth and came from God; neither came I of myself, but he sent me.
Jhn 8:43 Why do ye not understand my speech? [even] because ye cannot hear my word.
Jhn 8:44 Ye are of [your] father the devil, and the lusts of your father ye will do. He was a murderer from the beginning, and abode not in the truth, because there is no truth in him. When he speaketh a lie, he speaketh of his own: for he is a liar, and the father of it.
Jhn 8:45 And because I tell [you] the truth,ye believe me not.
Jhn 8:46 Which of you convinceth me of sin? And if I say the truth, why do ye not believe me?
Jhn 8:47 He that is of God heareth God's words: ye therefore hear [them] not, because ye are not of God.
I would be interested to see your citation where it says "in a general way" all men are the children of God