The “idea” infants needed to be baptized, in part, was because they believed circumcision, the OT shadow of the spiritual baptism, told them to do it. Ritual was a major part of religion at the time. Did the Apostles agree this meant sins remitted and salvation for infants? John the Baptist, Jesus, and the Apostles didn't mention infant baptism, and as far as we know did not practice it. The Apostles fought and spoke against philosophies, Gnosticism, and ‘their rival mystery religions’ of the day in their epistles and said when they departed wolves would bring damnable heresies, not sparing the flock, speaking perverse things, drawing away disciples and overthrowing the faith of some.
I’m not going to write an essay on Church History, because it speaks for itself on this subject if you take the time to learn it. Christianity didn’t corner the market on baptism or mystery religions. Other religions much older than Christianity baptized and because of similarities Christianity was considered another mystery religion. The world of the early Church was one of paganism, ritual, and the need to 'do something' and contribute to their religion, and religious ‘competition’. The second century Church was different than the first, and the third different than the second, and so on, all getting further away from Scripture because different religions and cultures influence each other when they mingle. Many don’t attempt to understand what this was like, and unfortunately interpret scripture without the historical and cultural lens. That converted believers baptized their infants doesn’t assume Augustinian/Reformed/Calvinist original sin/sin nature, but rather reconciliation to the Creator, law, and ritual. Did some, or even many, baptize their infants because of an Augustinian/Reformed/Calvinist original sin/sin nature view of sorts? Since they were not free of bad influence, what do you think? Does this mean they were right on either account? Of course not. Some did it because of “law” and ritual, or because they believed they had the authority to retain and remit sins, even for an infant that could not believe or not believe for themselves. Are these correct views? No they’re not.
Since there’s no Scriptural or Apostolic reason they came to that conclusion (baptize infants) and since we know the history and culture, what should we conclude?
Did Tertullian come up with this on his own through scripture? How come no one else did before him? Are you telling me the Apostles explained and passed on such an important doctrine as this and it was forgotten and ignored until Tertullian and Origen got the wheel of original sin/sin nature rolling again over 100 years later? I’m not buying it! The doctrine is the result of man not understanding scripture and being influenced by other philosophies and religions of their day.