So, I am not a paeodobaptist, but I am compelled to comment on this only because I don’t think you’re giving the paedobaptist position a fair portrayal. I am sure that there are many parishioners of reformed churches – mothers and fathers alike – who baptize their babies from a position of sub-conscious, sentimental, sacramentalist ignorance. But that doesn’t mean that they fairly represent paedobaptist theology. We could erect a similar strawman of credobaptism by pointing to the ignorance of countless teenagers being baptized at summer camp after having an emotional experience around a bonfire. And for every ignorant paedobaptist parent, there are just as many baptist parents who rush to get their children baptized upon the first faint glimmer of hope that their child is a theist. So let’s just make sure if we’re going to attack paedobaptism that we target it on its theological grounds, not just on the grounds of its abuses.
I believe your appeal to the Ethiopian eunuch actually dispels your argument rather than affirming it. You assert that through baptism the eunuch was placed under church authority. But exactly which visible church was he joining himself to after he was baptized? What elders was he placing himself under? What congregation was he being joined to as a member? None. He believed, was baptized, and then drove back to Ethiopia. Baptism, then, depicts a person’s birth into Christ. It does not constitute entry into a visible, local church.
Communion, by contrast, is explicitly relational in the Scriptures among the assembled body of believers. It is something we do with one another, as part of one another. When we partake of the Lord’s table, we partake together. And it is in this communion that our togetherness is represented. Moreover, it is the exclusion from this communion that God has instructed us to use as the vehicle of church discipline (1 Corinthians 5:11). Therefore, “church authority,” as you’ve called it, is bound in communion, not in baptism.
Perhaps your response to me would be to appeal to the authority of the “universal church.” But if that’s the case, my appeal to you would be to go back to the Scriptures and see that all ecclesiastical authority flows from the framework of local, visible churches. The universal church is not a coherent, visible entity. It is a reality that will only be realized on that great day when all the saints appear together in glory. Until then, I’m afraid the “universal church” is just a scapegoat for rebellion against God’s good order.