Character mistake: Donatello is running a rather verbose program analyzing the scepter. On screen the text mentions the "time-space CONTINIUM" rather than "continuum." (00:17:45)
Character mistake: Pretending to be a witch, April says that the song the walkman played was obtained because "I shrunk four incredibly cool guys and I locked them in there." The Barrio Boyzz were five at the song's release. Did she let one Barrio boy go? Was one of the Barrio Boyzz uncool? (00:16:40)
Answer: Michelangelo wants to stay behind for the same reasons as Raphael: because in that time and place, they are well respected, even revered as heroes. They can live openly and not be banished to sewers or subway stations.
LorgSkyegon
Mikey didn't want to stay behind because of Yoshi but because of Mitsu. Raphael even points out that she's the reason he doesn't want to return. When Mikey says they should stay, Raph responds, "Now, forget about Mitsu and give me the scepter."