I don't think Yuu could have used the time powers even if his eye hadn't been fucked up.

I'm pretty sure the time travel power worked kind of like the future gadget #8 in Steins;Gate, where it only sends the users current memories back in time to their earlier selves and not their current selves back in time. This is evidenced in the show as we see Yuu's brother revert to his younger body when he uses the power. So from that we can assume that this also happened to Yuu when he used the power to save his sister. That would mean that in this body he had not yet taken the power to time travel and therefore couldn't use it regardless of his eye.

Also using this you can retcon out the shit final ability and the bullshit luck of his brother finding him at the last second. Yuu's brother said that he would take care of all of the ability wielders in Japan so we can assume that the scientists had found a cure or something to suppress the powers, so Yuu's brother starts curing all of the Japanese ability wielders, but since he and everybody else assumes that Yuu has taken his power they don't bother wasting any of the cure on him.

(He wouldn't have been able to determine whether he still had the power anyway since 1. he was blind and couldn't use it and 2. Kumagami wouldn't have been able to detect it since his ability worked were he would randomly become aware of the location and ability of people with powers that his power hadn't identified before, otherwise he would have been constantly re-detecting power users in the school and base.)

So everyone in japan except Yuu's brother is cured and Yuu is the only person with the ability to locate powers left. Yuu also has the ability to see what the power he plunders does when he takes it and this is shown after he takes some of the more important powers. So at this point the rumours about the "one eyed reaper" have spread across the world and Yuu's older brother catches wind of that, using information from the network of ability wielders and the vast amount of money he has at his disposal he gets a rough idea of where his brother is and sets out to bring him back.

Yuu takes the power from the random kid in Beijing at the start of the scene and that is the actual last power holder except for his brother which is the last dot on the map that he sees. The little girl doesn't actually have a power she's just extremely brave, thats why Yuu doesn't see a power after plundering her, he just assumes that its courage since she started freaking out after he took the power. She actually started freaking out because she didn't expect the guy to actually shoot her and a crossbow bolt just started magically floating in front of her face right before it would have killed her. He never checks the map to see if she was the last ability user. So the reason his brother was so close and came just in the nick of time was because his ability was still present and Yuu was following the last red dot right to his location.