That’s true. It’s also true, save for Jesus’s birth and being at the temple at age 12, we don’t have much background about His life before His ministry starts.
Whereas Elijah and Melchizedek come out of nowhere. And John is compared to Elijah and Jesus is of the order of Melchizedek (Hebrews 5:6,10 and Psalm 110:4).
Your analogies are good. Our Sunday evening worship was recently going through Luke 1, and that’s why I responded.