Plot hole: The plot takes a turn for the nonsensical when the Turtles find the Scepter and Mitzu walks off. With no explanation (she is the best fighter of the resistance) Whit captures her. There's no explanation on how he did it with such ease, how did he happen to be at the right place at the right time since nothing of what happened was under his control, and even where this 'right place' was, because he is riding towards the village (and the Turtles themselves) with her - just how far did Mitsu wander during Grandpa's very short apology to be captured by someone who gallops to the village to show her off as bargain chip? (01:05:45)
Plot hole: It is explicitly said, with a mathematical equation, that time travel requires subjects of equal mass to swap places. Eidan Hanzei has a good 4 inches over Paige Turco, and the 4 honor guards don't have the same build between themselves, and surely not weight the same as the big and muscular turtles, who also have their shell adding further to the mix.
Plot hole: Michelangelo ends up in the forest completely by chance because he can't ride and his steed dragged him away from battle. Not only Mitsu is there in wait exactly on the tree branch he'll pass by, but also a bunch of villagers are standing by with a cart to fetch him. There's no possible reason why the leader of the rebellion would be lurking in wait setting up a trap in that random spot in the woods, away from battle, where no Norinaga soldier would be. (00:25:30)
Plot hole: April and Kenshin switch place in time as they were - by some extraordinary coincidence - holding the scepter "at the same time" (if the concept makes any sense) in the exact same pose, with a switch that takes several second of intense lightning storms and 'tornado weather' as one of the Turtles put it. Here there is the assumption that 4 priests would be around the scepter exactly at the same time, and when the Turtles do switch with Norinaga's elite soldiers instead they certainly were in completely different poses, far away enough to ride horses, who did not get spooked one bit by the lightnings and sudden winds but rode as if nothing happened.
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."