I think Rey feeling unworthy of Luke’s saber plays well into her feeling like she’s not able to live up to Luke’s legend and being the last Jedi, which is a good character arc, especially with the added puppet scene.
This would be great if there were some form of payoff but Leia gives it straight back in the next scene so it kinda makes it pointless.
The payoff is the climax of the movie, where she becomes “all the Jedi”. Suggestions of her feelings of unworthiness are also still present elsewhere in the movie because she leaves herself stranded on Ahch-To.
On the topic of Luke’s line on Leia: “Always in motion is the future”
It was unwise for Leia to give up her training because she sensed the death of her son if she were a Jedi, sure. It was also unwise for Anakin to become a Sith because he sensed the death of his wife. But since these two characters are related, it makes perfect sense IMO that they would make a similar mistake to save a loved one and fail in the process.