The years old statement is actually a big machlokos achronim, there are some that say that after twenty you do get an oinesh for what you did from thirteen.
We are all forgetting what Chazal say in פסחים דף נ’, לעולם יעסוק אדם בתורה ובמצות שלא לשמה, שמתוך שלא לשמה באה לשמהת the של”ה says that although it is actually an aveirah to learn שלא לשמה, it is impossible to achieve lshmah without first learning shlo l’shma.
This is all besides the point for there are two seperate discussions going on here, which for some reason is being jumbled together. The first point is What is the ultimate goal, and that is l’shma, not for schar. The second point is: How do we reach such a level, and that depends on the person. For some its schar, and for some it’s the oinesh.