This question is very interesting and, to a certain extent, it matches the reality of the first chapters of Acts. However, some nuance is needed.
Preaching aimed at the unconverted
In the early days, the apostles indeed devote a large part of their activity to the public proclamation of the Gospel. Peter preaches to the crowds at Pentecost (Acts 2.14-40), then in the Temple after the (Acts 3.12-26). The apostles teach daily in the Temple and from house to house (Acts 5.42). Their goal at that time is to convince Israel that Jesus is the promised Messiah.
This priority is understandable: the Church is still small, the eyewitnesses of the resurrection are present, and Jesus’ command is to be his witnesses (Acts 1.8). Christianity is above all a missionary movement.
But believers are not neglected
Yet Luke emphasizes from the start:
“And they continued steadfastly in the apostles’ doctrine and fellowship, in the breaking of bread, and in prayers” (Acts 2.42).
This remark shows that new converts received regular teaching. Luke reports practically nothing of its content, because his interest is mainly historical and missionary. He recounts the progress of the Gospel rather than describing the inner life of the assemblies.
A probably significant doctrinal deepening
It is hard to imagine that three thousand converts (Acts 2.41), and then several thousand more (Acts 4.4), were left without instruction. The Twelve devoted a significant part of their time to teaching (Acts 6.2-4). They surely passed on the words of Jesus, his , the accounts of his life, and the interpretation of the Scriptures that he had given them (Luke 24.27, Luke 24.44-45).
It is even possible that this teaching ministry formed the basis of the traditions that would later be put in writing in the Gospels.
The later development
As the Churches become more numerous and more distant from the apostles, the need for deeper teaching clearly appears. The epistles of Paul, Peter, James, or John testify to a constant effort to strengthen believers, correct errors, and develop Christian doctrine (Ephesians 4.11-14; Colossians 1.28; 2 Peter 3.18).
Another explanation
The impression that the first chapters of Acts are essentially directed toward the unconverted also comes from Luke’s very project. His book is not a catechetical manual, but the account of the expansion of the Gospel. He is mainly interested in extraordinary events, missionary speeches, and the progress of the Church. He leaves in the shadow a large part of the daily life of the Christian communities.
Thus, from a historical point of view, it is probably an exaggeration to say that the first Christians gave priority to evangelism at the expense of spiritual deepening. It would be more accurate to say that Luke places the main emphasis on the outward mission, while he only glimpses the important work of inner building up accomplished by the apostles. In reality, the two dimensions seem to have coexisted from the very origins of the Church.