The following is part of a discussion with some friends about baptism, starting with whether we should use the term "paedobaptism" [paedo=infant/child].
You may have some experience with Roman Catholicism that I don't regarding the term paedo, in which case I bow to a sensitivity I am unaware of. Yet, the term paedo is not Roman property. As I understand it, it is not like the term "Mass," describing a uniquely Roman doctrine, but more like the term "Intercessory prayer." Both Rome and Protestants use the latter term, mean different things by it (prayer to saints comes to mind), but can both use it legitimately. The term Paedobaptism describes who is eligible to receive baptism; it does not describe all that Rome believes about baptism, even if they use the term. Just because Rome uses a term with regard to an issue they are wrong about doesn't mean we can't use the term. Other examples of the same problem would be "Eucharist," "justification," "confession." I agree it is good to emphasize the covenant in baptism, yet there are times to emphasize who is eligible for baptism, and paedo is an appropriate term to use to advocate infant baptism. If we only use the term covenant for baptism, we will be confused and talk past each other when a Presbyterian talks to a Reformed Baptist.
You are both explaining Calvin's doctrine of the sacraments, without realizing it, perhaps, and missing a critical element. As you've said, Baptism is more than an "empty" symbol - just a picture (anabaptists and Zwingli). But, as you've said, it doesn't save in itself (Rome). The middle ground held by Luther and Calvin (though they each had different opinions themselves!) is that the sacraments are a means God uses to save us. Protestants, always leaning away from Rome, often re-interpret the "grace" in means of grace as "a nudge in a more godly direction." But it means salvation. God uses the sacraments to save those who "use" them with faith. The critical element you are missing that Calvin relied heavily upon is the Holy Spirit. It is He who unites us to Christ, by our faith, in the sacraments. Faith receives Christ in the sacraments. Now, if the sacraments are never given (stillborn baby, example), then faith receives Christ directly from the Spirit. But normally God gives us outward, objective signs and seals of our reception of Christ, for our assurance and edification.
Evangelicals have a hard time with covenant, for two reasons: 1. because they see themselves as either "in" or "out" with God - saved or not. 2. We are individualists who don't see our corporate life as a church before God as important, or impacting the real, personal relationship we have with God. [Back to #1] But God deals with people on another level besides our converted status, namely, that of covenant. You can have faith in God and not yet be reckoned among His people, like Rahab or Ruth. Or you can be reckoned among God's people, without faith in Him, like Eli's sons, or maybe Saul, or Ahab. Evangelicals just brush aside the covenant aspect, figuring that salvation is all that really matters. But they have to contend with Paul in Romans 3:1-2. Even though covenant signs like circumcision and baptism don't guarantee your right standing with God, they are still of great advantage!
What's up with "Baptism now saves you" in 1 Peter 3? It is the same as in John 6, where Jesus says you have no life in you if you don't eat My flesh. This is sacramental language, assuming the outward sign of water and wine is united with the spiritual reality of Christ's saving presence, by the Holy Spirit. There are times when that union isn't there - a hypocrite partakes unworthily.
Unbaptized children: a 1 year old boy died last year, in our congregation, not having been baptized. This didn't dismay anyone. We know he is with the Lord. His parents don't need the external comfort of baptism to know this. We also need to bring our children to Jesus and apply God's promises to them. I believe baptism is like a wedding ring - an external sign of the covenant that is real, even without the ring. But this doesn't keep us from wearing rings, as reminders to ourselves, to our spouses, and to the world, of who we belong to. Children without that for the first 6-15 years of their lives are really missing something.
On Roman baptisms: our elders decided quite recently, after much discussion, to accept Roman Catholic baptisms. Our rationale was that we must allow fellow Christians the liberty to be wrong on doctrine, while not declaring them heretics and unbelievers. It is not our correct belief about justification that makes us Christians, but our simple trust in Christ. We are not justified by affirming justification by faith alone; rather, we are justified by faith alone. Just as an Arminian will go to heaven with the wrong idea about God's sovereignty in his salvation, so many Catholics will go to heaven with the wrong idea about having to do certain works to gain God's favor. Our faith wavers and has many weeds like this one attacking it, but as long as it is rooted in Christ, the smallest grain of faith will save (1 Cor 3:15). We can't necessarily tell what is saving faith, but God can (Rom 2:15). The error can be great, but as long as it doesn't touch the nature of who Christ or the Spirit is, it is not beyond the pale. Of course, some Roman Catholics may be making an idol of the Roman Church, which is another question, altogether. But Luther and others certainly taught Reformed believers to look to their Catholic baptisms for secondary assurance of their salvation, with little to no thought of inconsistency. The anabaptists were the ones who thought they had to rebaptize (anabaptist is greek for rebaptize); Luther and Calvin and Bucer and Melanchthon and Zwingli and most others didn't. The legitimacy of baptism doesn't depend on the moral or doctrinal purity of the one baptizing, so long as they are in the pale of basic Christian orthodoxy regarding who they are baptized into (Trinity).
Grace and peace, in Christ!