Well the Father provided the Son and He said He draws men as well. Jn 12The 2 references you had cited credit the Person of the Father as drawing men unto the Son so you cannot emphasize the power as one shared by the Son & the Spirit that is drawing men unto the Son. No exception since the only exception for Who draws men unto the Son is the Father as in no one can come to the Son except the Father draws him.
So I agree with you in that our believing in God is a work of God also but more specified as the exception is the Father as He is the only One that draws men unto the Son.
Indeed, the Father is the One that reveals the Son to even babes so they can believe in Jesus Christ.
Matthew 11:25 At that time Jesus answered and said, I thank thee, O Father, Lord of heaven and earth, because thou hast hid these things from the wise and prudent, and hast revealed them unto babes. 26 Even so, Father: for so it seemed good in thy sight. 27 All things are delivered unto me of my Father: and no man knoweth the Son, but the Father; neither knoweth any man the Father, save the Son, and he to whomsoever the Son will reveal him.
Matthew 11:25-27 testifies of this work being of the Father in regards to hiding the truth from some while revealing the truth unto others.