Your instincts are good. It is a tad weird. ‘Being outstanding’ is a little vague. ‘Finally’ hints that better results had been expected earlier. And its opposite wouldn’t be ‘bad’. It would be ‘average’ or ‘mediocre’. So your sentence doesn’t suggest the same meaning as what your explanation says. The highly touted youth disappointed fans in his first three seasons. Last year, though, his hitting started to stand out. He led the team in home runs and finished 3rd in the league in RBIs. He had been a star in the minors and expected the same in the majors. But he was injured in his rookie season and put up only mediocre numbers for several years. But he never gave up on himself and last season finally put up the outstanding numbers that he knew he was capable of.

It's not weird, but a bit cumbersome. "He had finally begun to shine." "Finally, he began to excel." "He was outstanding, at last!" "He had finally proven himself (to be) outstanding." "Finally, he excelled."

While the phrase "He finally started being outstanding" is grammatically correct, it might sound a bit awkward or redundant. Instead, you could say something like: "He has finally become outstanding." "Recently, he has been outstanding." "He has finally started performing outstandingly." These alternatives convey the idea that the baseball player has improved and is now excelling without the repetition of the word "outstanding."

It makes sense in English and you will be understood. You could also use: "he is finally living up to his potential"