I did it! I finally 100%-d the game!
This was my second playthrough when I tried 100%-ing it. I had most of the stray beads from my previous attempt. I don't know how, but I managed to beat Hayate's 3rd race and Kai on the same day in 15 minutes. I just tried to map Hayate's changed route, but I managed to instantly tackle him at the first time when he turned, on the exact same spot where I tackled him in his 2nd race a day before. After that I tried the Kai race, and I was only behind an inch when she entered the finish line. I tried again and somehow managed to beat her. I must've been blessed by Amaterasu herself (it was a Sunday), because I'm usually not very good in games where quick reaction times are needed. It seems like overcoming hard challenges always happen when you least expect it. So if you're struggling with one of the races, focus on trying without expecting anything, knowing that if you try enough times, you will eventually make it.