Would Paul Have Known Much About Jesus' Childhood?

I've argued elsewhere that Paul tells us more in his letters about Jesus' childhood than people typically suggest. But before we even get to that sort of content in his letters, how much sense would it make to think Paul was as unconcerned about or ignorant of Jesus' childhood as people often suggest?

I've written some posts about what the Old Testament anticipates concerning the childhood of the Messiah, such as here and here. Paul was influenced by the Old Testament and would have had some interest in Jesus' childhood accordingly.

He was in contact with and following the work of multiple members of Jesus' immediate family for multiple decades (Acts 15:13, 21:18, 1 Corinthians 9:5, 15:7, Galatians 1:19, 2:9). The same is true of his relationship with the Twelve (Acts 9:27-28, 15:7, 1 Corinthians 1:12, 9:5, 15:5, Galatians 1:18, 2:7-10). In other posts, I've discussed how a traditional Christian view of Jesus' childhood is reflected in many places in all four gospels (Matthew, Mark, Luke, John). Paul was closely associated with all four of the individuals those documents have traditionally been attributed to. In addition to what's cited above regarding Matthew and John, see the following regarding Mark and Luke: Acts 13:5, 15:36-40, 16:10, Colossians 4:10-14, 2 Timothy 4:11, Philemon 24. Even if all four traditional gospel authorship attributions were to be rejected, Paul's widespread traveling and frequent contact with messengers among the churches and in other circles make it highly likely that he would have been familiar with at least much of the material on Jesus' childhood that was circulating and eventually appeared in the gospels. He also would have encountered, or had a lot of potential to have encountered, some figures who don't get much attention in modern discussions about these contexts, but would have had relevant information (e.g., Manaen in Acts 13:1).

Notice that even if we were to grant some skeptical scenarios for the sake of argument, some of what I'm appealing to would remain valid. Think of the third gospel, for example. The evidence suggests it was written by Luke, a companion of Paul, but let's assume for a moment that it wasn't. Why should we think it wasn't written by some other companion of Paul, then, given the evidence of the "we" passages in Acts? And if it wasn't written by any companion of Paul, why think it wasn't at least written by a segment of early Christianity that thought highly of Paul and, therefore, was highly influenced by him (which wouldn't require agreement with him on every issue)? The same line of reasoning can be applied to 1 Timothy and its citation of Luke's gospel as scripture (in 5:18). Even if we were to reject the Pauline authorship of 1 Timothy, why go even further by rejecting the document's reflection of Pauline thought and, more specifically, the perception at the time that Luke's gospel was circulating during Paul's lifetime and was viewed highly by Paul? More could be said, but what I've already brought up is enough to make my point. You can't get around the implications the gospel of Luke has for Paul's circumstances (e.g., what information he likely had access to, what views of Jesus' childhood he likely held) simply by doing something like denying the Lukan authorship of the third gospel or denying the Pauline authorship of 1 Timothy.

It seems highly likely that Paul had substantial interest in Jesus' childhood and access to reliable information on the subject. He probably held a view similar to what we see in the gospels.