“Do it” is certainly something Janeway says a lot, but it isn’t specifically her command to engage the warp drive, nor do I believe it was something that the writers room intended to be a phrase she put any thought into, but rather something they felt suited her character.
For that matter, I don’t think Picard was intended to be a character that put any forethought into his *“Engage,” and, “Make it so,” catchphrases.
The idea that every character needs to have a “thing” irks me probably more than it should. Saru trying to come up with a phrase suited the character to an extent because he was anxious and trying to project a certain authority, and also he wasn’t work on it on the bridge. I’ll even forgive Captain Freeman her “Warp me!” because it’s part of her character that she’s trying to stand out and get more recognition. But here, when the characters are ostensibly on a rescue mission they felt was important enough to steal the Enterprise, they still take a pause to banter about the need for a “thing.” Nah, this is easily the worst trope of modern Trek.