Personally, I think we'd be better off working towards a world where we didn't need a concept of gender and just biological sex was enough to determine who we are.
Yes, we should get rid of gender, no, sex should not "determine who we are". Sex shouldn't "determine" anything. That's how we got to that whole gender nonsense in the first place. People ought to be able to define their own personalities without constructing gender roles around a simple fact like sex.
Therefore there will be children who look like boys starting their periods
No. There won't. There will be children, who look like girls, and may or may not "identify" as boys, starting their periods.
Young girls may think that cutting their hair short and not wearing pink clothes will make them look like boys, but once puberty starts, almost all of them will be clearly recognizable as girls. (And before that, they may well look like children. You wouldn't say a baby girl who doesn't have a bow glued to her head looks "like a boy" would you?)