The gift an infant has is the gift of faithful parents who promise to bring up the child in the nurture of the Lord - a very great gift. But there is no need to claim that the infant has "saving faith" while still an infant. It is sufficient to observe that God included infants in the covenant he made with Israel when they left Egypt.
The holy scriptures tell us what baptism means in passages that has previously been cited and quoted.
I see baptism in Jesus discussion with Nicodemus
John 3:3-8 Jesus answered: In all truth I tell you, no one can see the kingdom of God without being born from above. [4] Nicodemus said, 'How can anyone who is already old be born? Is it possible to go back into the womb again and be born?' [5] Jesus replied: In all truth I tell you, no one can enter the kingdom of God without being born through water and the Spirit; [6] what is born of human nature is human; what is born of the Spirit is spirit. [7] Do not be surprised when I say: You must be born from above. [8] The wind blows where it pleases; you can hear its sound, but you cannot tell where it comes from or where it is going. So it is with everyone who is born of the Spirit.
I see baptism in Paul's words about salvation
Titus 3:3-7 There was a time when we too were ignorant, disobedient and misled and enslaved by different passions and dissipations; we lived then in wickedness and malice, hating each other and hateful ourselves. [4] But when the kindness and love of God our Saviour for humanity were revealed, [5] it was not because of any upright actions we had done ourselves; it was for no reason except his own faithful love that he saved us, by means of the cleansing water of rebirth and renewal in the Holy Spirit [6] which he has so generously poured over us through Jesus Christ our Saviour; [7] so that, justified by his grace, we should become heirs in hope of eternal life.
I see baptism in Paul's discussion about union with Jesus Christ
Romans 6:1-14 What should we say then? Should we remain in sin so that grace may be given the more fully? [2] Out of the question! We have died to sin; how could we go on living in it? [3] You cannot have forgotten that all of us, when we were baptised into Christ Jesus, were baptised into his death. [4] So by our baptism into his death we were buried with him, so that as Christ was raised from the dead by the Father's glorious power, we too should begin living a new life. [5] If we have been joined to him by dying a death like his, so we shall be by a resurrection like his; [6] realising that our former self was crucified with him, so that the self which belonged to sin should be destroyed and we should be freed from the slavery of sin. [7] Someone who has died, of course, no longer has to answer for sin. [8] But we believe that, if we died with Christ, then we shall live with him too. [9] We know that Christ has been raised from the dead and will never die again. Death has no power over him any more. [10] For by dying, he is dead to sin once and for all, and now the life that he lives is life with God. [11] In the same way, you must see yourselves as being dead to sin but alive for God in Christ Jesus. [12] That is why you must not allow sin to reign over your mortal bodies and make you obey their desires; [13] or give any parts of your bodies over to sin to be used as instruments of evil. Instead, give yourselves to God, as people brought to life from the dead, and give every part of your bodies to God to be instruments of uprightness; [14] and then sin will no longer have any power over you -- you are living not under law, but under grace.
I also see baptism in Peter's discussion about salvation
1 Peter 3:18-22 Christ himself died once and for all for sins, the upright for the sake of the guilty, to lead us to God. In the body he was put to death, in the spirit he was raised to life, [19] and, in the spirit, he went to preach to the spirits in prison. [20] They refused to believe long ago, while God patiently waited to receive them, in Noah's time when the ark was being built. In it only a few, that is eight souls, were saved through water. [21] It is the baptism corresponding to this water which saves you now -- not the washing off of physical dirt but the pledge of a good conscience given to God through the resurrection of Jesus Christ, [22] who has entered heaven and is at God's right hand, with angels, ruling forces and powers subject to him.
Some may say that the references in the first two passages are to something other than baptism.
Since this thread is about the meaning of baptism I quote the passages above as instances of the holy scriptures presenting perspectives on the meaning of baptism. The question of who may be baptised is not the same as asking what baptism means.
The proper candidates for baptism is a matter disputed by Anabaptists and Baptists and those denominations and non-denominational groups that claim baptism is to be administered only to those who make a credible profession of faith prior to being baptised. The rest of the Christian community regards baptism as rightly administered to believers and their children. If this matter were decided by vote then paedobaptism would be received as right and if the matter is judged by the practise of the ancient church then paedobaptism would be received as right if the matter is decided by what the holy scriptures say then we have a dispute because those sharing an Anabaptist perspective say one thing and those who share the perspective of the rest of the Christian community say a different thing. Deciding what is right by scripture's words alone hasn't worked for almost 500 years. It isn't going to work now either.