One I've had to do super often is injecting a name back in a sentence. Why say

Mary said the following about Jane: "She went to the store today."

when I could say

Mary said, "[Jane] went to the store today."

I mean, I could just paraphrase Mary and do away with the quotation marks and brackets entirely, but when I am trying to prove something (primarily that I'm not talking out of my ass) I like quotes because you can easily just take it as direct evidence, an exact citation of what the other person said that you can use as evidence yourself, instead of a paraphrase by some random person whose reasoning and motives you do not trust.

Of course, that doesn't get into how people can manipulate quotes and take them out of context, or even just straight up write something in quotation marks that was never said, but…