Well, first of all, the Pope is not obliged to publish an exhortation. This is a very recent practice that began after the Second African Council. He is not obliged to do this. Therefore, it is not a traditional practice to publish apostolic exhortations or post-synodal apostolic exhortations. It depends on the circumstances and context.
The Synod is only a consultative body, according to the concept of the so-called Synod of Bishops established after the Second Vatican Council. It is not a body that communicates binding decisions.
The problem here is that Pope Francis declared the final document of this so-called Synod as part of his papal Magisterium. Of course, a pope can take a text and then publish it under his authority and declare it part of his Magisterium. But in this case, the document was voted on formally by lay people who had the same voting rights as bishops, who are the Magisterium.
Therefore, it creates confusion. It gives the impression that lay people, in some way, also belong to and share in the teaching office of the Church. This is not Catholic. It reflects a Protestant understanding, not a Catholic one.
Lay people are not members of the Magisterium and cannot formally share in the teaching task of the Church. Of course, lay people in families, in catechism classes, and in schools must transmit the faith and teach it to children. They serve as catechism teachers and witnesses to the faith in the world. But they do not formally belong to the Magisterium of the Church.
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