This might be a difficult question to answer:
What do the written scriptures say is the purpose of baptism in Jesus's name? Specifically, what do the texts say?
The usage is not perfectly explained, however there are illustrations, usages and allusions. I suggest starting with John 15 with the famous "I am the vine, and you are the branches" plus two other passages: Ezekiel 17 the parable of the 2 eagles and the vine plus a cedar and Matthew 13:19-20 the explanation of the sower and the seed. The sower and the seed parable probably borrows from Ezekiel 17 and is related. Its got the snatching eagle, whereas Jesus parable has the deceiver who steals the seed. There are some parallels to the stories.
Ezekiel plainly states that the eagle story is a parable. The point of it is similar to what John the Baptist preaches: that the high will be made low and the low will be raised up. John preaches this and quotes from Isaiah's passage about a Voice calling for the low places to be raised and the high places made low. The new baptized people are the by product of this raising and lowering. The imagery is that it is a path in the wilderness, however you can easily also see a stream or a vine running parallel to said path. They are shared meanings perhaps. In the parable of the two eagles, the new baptized are the birds 'Of every kind' which nest in the cedar. Jesus through Matthew brings this no doubt popular parable to recall with his own parable and shows a unity between John the Baptist's preaching and Jesus preaching.
Jesus says he is this 'Vine' that we all must be in, and he is also the soil which produces good fruit. The parable has other levels, because anyone can be good soil, and each person receives the Logos. Its a parable with facets. The symbol of the vine has many sources, however look at Genesis 49:11 and 49:22. Its often referred to as 'The branch' which prophets like to refer to when they are talking about the direction that Israel must go as well as a symbol of other things which I'll leave out partly for brevity but mostly because I'm not the person who'd know them all.
How this ties into being baptized into his name is that he is the vine, aka the head of a family. He is also the final cedar with birds of all kinds in its branches (Not all eagles but all kinds) when we place him into the eagles parable which carries that same message of John the Baptist, and he is the good soil in his parable about seed.
Baptism then is joining a family but not a physical family or a family that is defined through adoption or human inheritance but inheritance from a heavenly Father, and the Logos gives the person the power (according to
John 1) to become a son of God, not requiring adoption by a Jewish family or some special teaching method or school of thought or system. So 'Birds of all kinds' not 'Birds of a feather', twelve very different disciples, joining this choicest vine, this most fruitful vine not by the will of any person, not talked into it or married into it or joined to it for social advantages. It is a very mixed, very diverse family that one is supposedly baptized into.
From this we can guess at the purpose, but Matthew recounts that its 'Glory reserved for God in the highest heaven, Peace on Earth, goodwill toward men' (slight paraphrase to emphasize that the glory is reserved for God only). The glory here is the essence of truth, and its shining from heaven not from the ground is what enables people of different opinions to be baptized into the same vine. It is because glory belongs to God in heaven and shines on us, much like we are a stained glass artwork and the light shines through it onto a floor somewhat distorted in the process but still beautiful and bright.