Your full question is a good one: “So why does Jesus say no one knows the Son except the Father, when he also says people can know him, the Son?” The answer lies in appreciating the different levels of “knowing” that there are in the Bible.
There is carnal ‘knowledge’ as happens in sexual intercourse (Matthew 1:25). This has got absolutely nothing to do with what Jesus meant, even though such an act is said by God to make the two "one flesh". Yet there is divine knowledge of the life growing in the womb, which is incredibly intimate but only known by God:
"Before I formed thee in the belly I knew thee; and before thou camest
forth out of the womb I sanctified thee, I ordained thee a prophet
unto the nations" Jeremiah 1:5 K.J.V.
That is reminiscent of the wording used to describe the apostle Paul's calling to be minister of the gospel to the nations (Galatians 1:15). In both cases, the humans whom God knew and ordained before birth had no knowledge of God until years later, and even so, they were grown men before God's elective calling was understood and obeyed by them. Then they began to "know" God in a particular way, but not mere head-knowledge. That could be obtained by prophet and religious leader alike by studying the written scriptures. Yet Jesus castigated them for merely searching the scriptures without coming to him to have life (John 5:39-40 K.J.V.). They prided themselves on their intellectual grasp, and memory of, the scriptures. Oh, they knew lots about God, but they didn't know God as Jesus knew God.
How did Jesus know God? With a divine intimacy that no mere human has. That is why Paul explained that God separated him from his mother's womb "to reveal his Son in me". By divine revelation, he was stopped in his self-important tracks to have Christ revealed to him. He did not know who the glorious one was - "Who art thou, Lord?" - and the Son revealed the truth that gives eternal life, which enabled Paul to then know God as Father, and Christ as the Son.
Sadly, there are always people who think that study of the scriptures will enable them to know God and Christ, but they think that means head-knowledge; knowing things about God and Christ. No. Never. That is why a translation such as this is so gross: "This means everlasting life, their taking in knowledge of you, the only true God, and of the one whom you sent forth, Jesus Christ" (John 17:3 KIT). The ones who produced that version get all their followers to do rigorous study about the scriptures, yet 99% believe that they will never even see Christ nor ever be with him. Compare that with what the apostle Peter discovered in Matthew 16:16-17. Jesus rejoiced that his heavenly Father had revealed that divine knowledge to Peter.
So there we have two classic examples of the need for heaven to break through into a person's understanding before they can know God and Christ in a saving way; Paul had Christ's direct revelation; Peter had the Father's direct revealing. That's how both statements are true, and work together; "No one knows the Son except the Father, and no one knows the Father except the Son and those to whom the Son chooses to reveal him" (as your translation of Matthew 11:27 puts it.)
When we start with what God has already revealed to us, we should be spared stumbling around what humans think. Yes, there was human inability to grasp the enormity of just who Jesus was, and Jesus must often have sighed and looked disappointed with them. Yet Jesus told them, and us, plainly, that we must come to him, personally, in faith, if ever we are to know his Father in the only way that can save us from mere knowledge.
In the John 10:14 text you mention Jesus speaking of those who recognise him as their shepherd. They do so because they know Jesus gave his life for them, the sheep, so that when he speaks, his voice is recognised by them as the voice of their shepherd, unlike false shepherds who abandon the flock. They know Jesus is always with them, guiding them (even now he is in glory in heaven).
The knowledge the Father and the Son have is unique, due to their unique relationship. That is what accounts for what Jesus was driving at. Of course, until he arose in triumph from the grave, they were lacking the full, divine revelation of the Son being declared to be the Son of God by his resurrection - Romans 1:3-4. Then the full impact of that (and, hence, Peter's declaration pre-resurrection, of Jesus being the Christ, the Son of God) came home to stay. Then believers had the indwelling Holy Spirit uniting them to their risen Lord. Then believers will go to be with their Lord in heavenly glory, the Holy Spirit bearing witness with them even now of this assurance Christ's resurrection gives them (Romans 8:1-27).
The "knowing them" that you ask about in Matthew 11:27 is not 'to know', ginosko; it is not 'known' - gnostos; it is not 'to see, have seen, known' - oioa oida; it is not 'to understand' - epistimai; it is 'to know fully about' - epiginosko. Just as "the fullness of deity" [i.e. the Godhead] dwells bodily in Christ (Colossians 2:9), so we are told in the next verse that those who belong to him "are complete in him" as he is their head. That is the sense in which Jesus spoke in Matthew 11:27. Because "all things are delivered unto me of the Father", that includes all believers who belong to him by faith. That is what brings them into a knowledge of the Father and the Son that goes beyond anything any human can attain to, for it requires the Father and the Son, in one Holy Spirit, to reveal such wonders to them.