OP: I want to know if truly Jesus want to found a church
Yes, Jesus tells Peter in Matthew 16:
And I tell you that you are Peter, and on this rock I will build my church, and the gates of Hades will not overcome it.
The OT temple is a type of the NT church. It foreshadows the church. The church is the perfect fulfillment of the temple. The church is the body of Christ, John 2:
19Jesus answered them, “Destroy this temple, and I will raise it again in three days.”
20They replied, “It has taken forty-six years to build this temple, and you are going to raise it in three days?” 21But the temple he had spoken of was his body.
This is the spiritual reality. Paul elaborates on this concept in
For the husband is the head of the wife even as Christ is the head of the church, his body, and is himself its Savior.
OP: In the above verse we have Christ being the head of the church, but how does it differ from synagogue (if there is a true difference).
Do not confuse synagogues with the temple, nor the local churches with the body of Christ. Synagogues and local churches are physical buildings that house attendants.
Paul is not talking about physical buildings.
And he put all things under his feet and gave him as head over all things to the church, which is his body, the fullness of him who fills all in all. (ESV)
OP: However isn't it more a spiritual concept of church in this case, for the institutions may fail, and also the humans that are part of it?
Precisely. Paul continues, Ephesian 2:
14For he himself is our peace, who has made the two groups one and has destroyed the barrier, the dividing wall of hostility, 15by setting aside in his flesh the law with its commands and regulations. His purpose was to create in himself one new humanity out of the two, thus making peace,
Paul was a Pharisee. He attended synagogues all his early life. According to him, the temple with its associated synagogues was to be subsumed into the one new church.
16 and in one body to reconcile both of them to God through the cross, by which he put to death their hostility. 17He came and preached peace to you who were far away and peace to those who were near. 18For through him we both have access to the Father by one Spirit.
19 Consequently, you are no longer foreigners and strangers, but fellow citizens with God’s people and also members of his household, 20built on the foundation of the apostles and prophets, with Christ Jesus himself as the chief cornerstone. 21 In him the whole building is joined together
and rises to become a holy temple in the Lord. 22And in him you too are being built together to become a dwelling in which God lives by his Spirit.
The temple has been fulfilled in the body of Christ in the spiritual reality, regardless of where you are. We are united in the body of Christ.