Given what Luke learned about the fall of the Jedi, it’d make sense to toss away the Youngling Killer 9000 like garbage.
I wish they had explored that further as it may have actually made me appreciate the idea behind Luke tossing it. It could work as Luke’s second lesson. He tells Rey about the fall of the Jedi and proceeds to tell her about the terrible acts the lightsaber has been part of in galactic history. He shows hesitation but gifts her his green one. She reluctantly accepts and they exchange lightsabers. She tells him that she’ll build her own and will return it back to him. He smiles slightly.
Kylo/Ben sees she has it and gets flustered. He proceeds to tell her his account of what happened with Luke. She becomes more hesitant to be wielding the green one but proceeds to the mirror cave as normal and they have their moment as he can look pass it as his feelings for her are greater.
Luke seems to have come around and is open to teaching her how to build a new lightsaber for his third lesson as a change of mind to his original plans as he has parts to build one. He even sees possiblity in her wielding a double blade if they combine the green and blue together or use the kyber Crystals in both. However as soon as he sees her with Kylo/Ben he gets angry again and eventually he tells her the third account of what really happened. This time she reaches out with his green lightsaber but he doesn’t accept and reluctantly tells her she will need it if she’s going through with her plan.
Kylo/Ben mentions in the lift that he can’t believe she has the lightsaber but everything continues as seen in the film except he uses the green lightsaber to defeat the guard. He comes around to keeping it for himself as a reminder of Luke and that it saved his life. He fights Rey for it but it snaps in half.
This also adds to when Luke uses the blue one on Crait. It helps create the further illusion that he is on the planet and ultimately he reclaims it as a symbol of hope.