Hi sasushi! It's hard to generalise what an author is trying to do across ALL persuasive texts, but there are certain generalisations we can make when an author repeats something. The repetition of anything (including tone) is meant to reinforce certain ideas, concepts or perspectives. So if an author returns to the same 'casual, relatable tone', it is probably to reinforce the idea that the author is a relatable person to their audience, which achieves a certain purpose, eg that so-and-so is an out of touch politician; that XYZ solution is the common sense approach, etc.
When an author bookends their piece with something (that is, start and begin their piece with a tone, idea, anecdote, etc), we can generally assume they are trying to reinforce an idea (as above) but also to bring a sense of closure to their overall argument as well. To start and end with the same thing will remind the audience of where the author began (argument-wise) and now that they've had the whole article to lay out their points, bring a new perspective to what they started with. Now what kind of closure and what that closure achieves - that will really depend on the article!
Hope that helps! 🙂
