As Jesus “grew and became strong,” being “filled with wisdom, and as “the favor of God was upon Him” (Luke 2:40), and as He began to live out the life of the Messiah within a community that generally believed that the Messiah would be the fleshly embodiment of their covenant making, creative, and providential God, He would undoubtedly have searched (or at least asked for assistance in searching) the collected Hebrew Scriptures for guidance in living according to messianic expectations. Naturally, Jesus was concerned with how to go about revealing the Creator God to His covenant people.
That said, it is quite right to here contemplate the “suffering servant” of Isaiah (despised, rejected, suffering pain, held in low esteem/honor, stricken, afflicted, pierced, crushed, punished, wounded, etc…). With knowledge of the way in which Jesus lived, the way He performed His ministry, the people with whom He surrounded Himself, the subversive challenge that He offered to the powers-that-be of the day (politically, religiously, socially, and culturally), and the way in which His choice of messianic presentation would lead to the way in which He would come to meet what was His presumed demise, it seems to be rather clear that He took the notion of Isaiah’s suffering servant as rather foundational for His mission.
Likewise, the authors of the Gospels, as they lived in the early days of the church that was growing and serving humanity and enduring threats and persecutions in the wake of Jesus’ Resurrection and ascension and the telling of that story, and as they did so in a time in which oral traditions (possibly informed by smaller written collections of Jesus’ words or deeds) concerning Jesus were circulating, sought to tell Jesus’ story within the framework of messianic (king of Israel/God in the flesh) conceptions (the suffering servant being one mode of telling) that were on offer within the Hebrew Scriptures.
Just as Jesus, to whom believers are able to look as the Creator God in the flesh, was concerned with properly revealing His God to His people, so too would His biographers be concerned with structuring their stories of Jesus with an eye and an ear to the very same thing. This notion is not necessarily talking about making Jesus fit within a certain mold, or telling His story so as to make it appear that He is the fulfillment of centuries-old prophecies, but rather, utilizing certain portions of the accepted Scriptures (and the self-understanding that shaped, was shaped by, and governed the reading of those Scriptures) that served to inform the people about the nature of their God, Jesus’ biographers, who believed themselves to be telling the story of the Lord and Master of the whole of the cosmos, shaped their presentations of the stories of Jesus so as to align with the nature-of-God-revealing statements that were part and parcel of the writings of Israel’s prophets, poets, and historians.
Surely, it is not at all difficult to understand that the Gospels offer an exceedingly small glimpse into the life of Jesus, so one does well to consider why it is that what was selected for recounting to and for the church communities, was in fact selected. Most likely, the structure of the presentation of events in the life of Jesus is taken for granted, with believers enjoying the narrative on offer without ever wondering why certain events follow one another within the Gospel presentation.
If one takes seriously the fact that Jesus lived as part of a community that took its story very seriously and understood itself according to the story recorded in its Scriptures, while also understanding themselves as the covenant people of the Creator God that routinely and often dramatically enters into history to act for or towards His people as they looked towards a great and seemingly final intervention, then this notion must be brought to a reading of the Gospels which then shapes the way that the remainder of the New Testament is read (though the Gospels and Acts would have been composed and circulated in written form after almost all of the rest of what would come to comprise the New Testament has been composed, though certainly the stories recorded in the Gospels and Acts circulated in what was likely a moderately controlled oral form before the written records were produced).