I know Super Mario 64 has these sort of runs with certain stars. All speedruns involve the IGT start to finish but you could calculate the time using the IGT or use an actual timer. I'm not a moderator so i don't make the final choices but from what i can see the possibility is there.

https://www.speedrun.com/Google_Doodle_ICC_Champions_Trophy_2017_Begins this game has already incorporated highscores into the website by using the times. But I still feel like a different website is a good idea (4 or 3).