Jesus as the Suffering Servant of Yahweh
The second reason why pais is better rendered as servant is because the context of Acts makes it clear that Jesus is being referred to as the Suffering Servant mentioned in the book of Isaiah, the One whom God appointed and empowered by his Spirit to bring justice and salvation to all the nations and to offer his life as an atoning for sins. These prophecies are commonly referred to as the Servant Songs and are four in number (cf. Isaiah 42:1-7; 49:1-10; 50:4-10; 52:13-53:12).
In fact, the Greek version of Isaiah known as the Septuagint (LXX) renders the Hebrew word for servant (ebed) as pais!
Note, for the instance, the following prophecy:
“Hearken to me, ye islands; and attend, ye Gentiles; after a long time it shall come to pass, saith the Lord: from my mother’s womb he has called my name: and he has made my mouth as a sharp sword, and he has hid me under the shadow of his hand; he has made me as a choice shaft, and he has hid me in his quiver; and said to me, Thou art my servant (doulos mou), O Israel, and in thee I will be glorified. Then I said, I have laboured in vain, I have given my strength for vanity and for nothing: therefore is my judgment with the Lord, and my labour before my God. And now, thus saith the Lord that formed me from the womb to be his own servant, to gather Jacob to him and Israel. I shall be gathered and glorified before the Lord, and my God shall be my strength. And he said to me, It is a great thing for thee to be called my servant (paida mou), to establish the tribes of Jacob, and to recover the dispersion of Israel: behold, I have given thee for the covenant of a race, for a light of the Gentiles, that thou shouldest be for salvation to the end of the earth. Thus saith the Lord that delivered thee, the God of Israel, Sanctify him that despises his life, him that is abhorred by the nations that are the servants of princes: kings shall behold him, and princes shall arise, and shall worship him, for the Lord’s sake: for the Holy One of Israel is faithful, and I have chosen thee. Thus saith the Lord, In an acceptable time have I heard thee, and in a day of salvation have I succored thee: and I have formed thee, and given thee for a covenant of the nations, to establish the earth, and to cause to inherit the desert heritages: saying to them that are in bonds, Go forth; and bidding them that are in darkness shew themselves. They shall be fed in all the ways, and in all the paths shall be their pasture. They shall not hunger, neither shall they thirst; neither shall the heat nor the sun smite them; but he that has mercy on them shall comfort them, and by fountains of waters shall he lead them.” Isaiah 49:1-10 LXX