Can anybody explain why 私 is a sixth grade kanji?
I find it funny that such a common word appears on the list so late.
This is because you don't have to write it in kanji even after you have become an adult. On BCCWJ, there are 65,182 examples of 私は, while the number of the examples of わたしは is 11,372. This means many adults choose to write わたし in hiragana even after learning its kanji. (I think formal documents tend to contain the kanji 私 more often.) By contrast, an educated adult almost never writes 学校 or 会社 in hiragana, so, in a sense, these kanji are more fundamental. 私 is not really an essential kanji for making sentences on a daily basis, and learning it in sixth grade is not too late. The same is true for あなた; even though it has kanji (貴方), not many people use it in day-to-day writings.