Your argument against paedobaptism is valid if and only if (1) people have free will AND (2) there is no power as the Word of God. The heart of the objection is "how can an infant prove -- with a free will -- that it believes in Jesus Christ in order to be baptized. "
Cornelius and his household in Acts 10 are said to fear God and give alms and be pious -- things which religions that believe in human free will hold as proofs of already being saved or of at the least 'earning' salvation if such works are continued.
The whole point of Acts 10 is that although Cornelius and his household already did these things, they were NOT saved (which makes Acts 10 a chapter used to disprove "works based" salvation). They were not yet saved, and did not receive the Holy Spirit until the instant they heard the Word of God that God as Word spoke through an already saved, an already born again individual: the Apostle Peter in this instance.
Act 10:44-47 While Peter was yet speaking these words the Holy
Spirit fell upon all those who were hearing the word. And the
faithful of the circumcision were astonished, as many as came with
Peter, that upon the nations also the gift of the Holy Spirit was
poured out: for they heard them speaking with tongues and magnifying
God. Then Peter answered, Can any one forbid water that these should
not be baptised, who have received the Holy Spirit as we also did ?
The sequence of events: Hearing the Word of God/ being born again first, baptism second.
Other places where God directly says, as differentiated from demonstrating as in Acts 10, that we are born again by hearing the Word of God:
1 Corinthians 4:15 For if ye should have ten thousand instructors in
Christ, yet not many fathers; for in Christ Jesus I have begotten
you through the glad tidings.
James 1:18 Of his own will begat he us with the word of truth,
that we should be a kind of firstfruits of his creatures.
1 Peter 1:22-25 Seeing ye have purified your souls in obeying the truth
through the Spirit unto unfeigned love of the brethren, see that ye
love one another with a pure heart fervently:
23 Being born again, not of corruptible seed, but of
incorruptible, by the word of God, which liveth and abideth for
24 For all flesh is as grass, and all the glory of man as
the flower of grass. The grass withereth, and the flower thereof
25 But the word of the Lord endureth for ever. And this
is the word which by the gospel is preached unto you.
The whole point of ACTS 10 is that the works Cornelius was already performing were acknowledged by God and yet were insufficient for salvation, because God requires we hear Him as speech/Word of God as distinct from human language which He confounded at the attempt to build the tower at Babel:
Gen 11:6-9 And the LORD said, Behold, the people is one, and they
have all one language; and this they begin to do: and now nothing will
be restrained from them, which they have imagined to do.
Go to, let
us go down, and there confound their language, that they may not
understand one another's speech.
So the LORD scattered them abroad
from thence upon the face of all the earth: and they left off to build
Therefore is the name of it called Babel; because the LORD
did there confound the language of all the earth: and from thence did
the LORD scatter them abroad upon the face of all the earth.
God doesn't bless with a language He has Himself profaned, a language that is not-God-as-Word/Christ.
As to infant baptism and the example of Cornelius's household:
We hear the Word of God first, receive the Holy Spirit simultaneous with that hearing, are born again as a result of that hearing, recreated as new creatures in Jesus Christ, receive the Holy Spirit along with it and THEN we are baptized. Infants have nothing to prove and quite frankly, nothing to prove it with: just as unsaved adults. Most objections against infant baptism are an implication that adults have some advantage that infants don't have even though Christ says explicitly
Mark 10:15 Verily I say unto you, Whosoever shall not receive the
kingdom of God as a little child, he shall not enter therein.
Infant baptism is based on the reliance of the power of the Word of God. Even in false religions that baptize children in the name of Christ, they say words in the ceremony prior to the water being splashed or "going under" in a direct attempt to mimic "Word of God first, baptism second". What baptism has anyone ever seen where no one says a word and they just use the water?
That doesn't mean that everyone who is baptized goes to heaven. False religions mimicking Christ / Word of God in human language don't work, don't birth anyone again, but they all pretend and that pretense, until God speaks to them -- as to Cornelius -- is their best honesty about reality. They are not "lying on purpose". They are pious, fear God and give alms but are not yet saved/born again. That means after they are saved, they need to be baptized for the first time the right way, not "again".
Saying that infants can't understand what is being said to them and thus can't "accept or reject" God's salvation is like saying God can't bless someone through me in English that only understands Russian. There is no such limitation in Christ. The reliance is on the power of the Word of God Himself, not human understanding. Lazarus couldn't "agree or disagree" with the Word of God that called him forth from death.
1Co 2:12-14 Now we have received, not the spirit of the world, but
the spirit which is of God; that we might know the things that are
freely given to us of God.
13 Which things also we speak, not in
the words which man's wisdom teacheth, but which the Holy Ghost
teacheth; comparing spiritual things with spiritual.
14 But the
natural man receiveth not the things of the Spirit of God: for they
are foolishness unto him: neither can he know them, because they are
Hope this helps.