Continuity mistake: When unconscious Michelangelo is lifted up and put on a cart, a man suddenly appears on the left.
Continuity mistake: When Michelangelo's horse is about to stop galloping, the wide shot shows his right hand lowered and his body bent downwards, but in the close-up his hand is raised. This swaps back and forth. Seemingly, the wide shots were of a puppet of some sort used for security reasons.
Continuity mistake: When the soldiers smash April's Walkman, pieces get scattered all over the place. In the close-up they're very close together.
Continuity mistake: At Norinaga castle, Kenshin walks towards the Lord and kneels down. The number of people around them change between shots. Also, the horse on the far right that started walking away is suddenly next to the soldier.
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."