I have a theory about where the idea of the "age of accountability" comes from. It's just a theory, I can't prove it. In part three of the Summa, Aquinas talks about confirmation among the sacraments. He rationalizes the existence of confirmation at age 13 or so, by claiming that people lack the moral faculties necessary to act as a responsible person within the Christian community up to that point. The symbolism of the anointing, he claims, is similar to when a person takes a fealty oath to a king or lord. They publically pledge themselves to actively fight or work for that lord or king. This is what happens in confirmation. The person now has the moral and intellectual agency to work to further their own growth in faith and the good of the Christian community. As children, they lacked capacities for either. God gives grace in order to augment and active this capacity.
Now, this is exactly what Baptists say about people who are under 13 or around about that stage in life. It is interesting that they just happen to choose the time in life that Aquinas mentions. They also seem to understand baptism in about the same way that Aquinas understands confirmation: it's about publically declaring your own active choice for Christ and willingness to be an active member of the Church community. It's broadly speaking, like a public fealty oath. The difference is that Aquinas thinks that confirmation conveys real grace (augmenting the created and uncreated grace received in baptism), whereas the Baptists consider baptism to be completely symbolic. Hence, what appears to be an entirely arbitrary scheme of the age of accountability is actually a holdover in the Baptists (without them actually realizing it) from the medieval theology of confirmation. What they've done is simply transfer the medieval Latin Church's rationale for confirmation over to baptism.
On a side note, confirmation goes back to about the 5th-8th centuries in the west. Aquinas (like Luther, and unlike Augustine and other medieval theologians) claims that all sacraments must have a dominical command, so he says that Christ promised confirmation to the disciples and that they received it on Pentecost. In any case, what happened was that a number of rites that the patristic Church associated with baptism (anointing with oil, etc.) were delayed until 13 or so. Usually the symbolism of these rites was tied up with having completed catechesis and entering the priesthood of all believer (i.e. the oil like the priestly oil of the OT). Since most of western Europe had been Christianized and there were very few adult converts for the first time, the majority people were getting baptized prior to having catechesis (i.e. as infants). So just as they delayed catechesis, so they broke off and delayed these extra rites associated with baptism and performed them when catechesis was completed- usually at around age 13. Hence, confirmation was invented.