Continuity mistake: When Michelangelo's rolled over by Mitsu after being knocked off his horse, his samurai mask is under his chin. A shot later, after the camera's on Mitsu, it's suddenly gone.
Continuity mistake: When Mitsu removes Raph's helmet, she grabs the lower edge of it. Shot changes and her hand is positioned higher.
Continuity mistake: April is moving the iron sewer door but from the opposite angle the door is lying still on the floor despite her pants of effort and despite the fact that a shot later the door has been moved.
Continuity mistake: When Michelangelo is about to be lifted on the cart Mitsu raises her hand and commands something to the soldiers. A shot later her hand is lowered.
Continuity mistake: When Michelangelo is about be knocked down by Mitsu, the galloping horse arrives at where the camera is and comes to a stop. When the shot changes he is galloping again.
Continuity mistake: When Michelangelo drops the sceptre it lies perpendicular to some green leaves. When Mitsu picks it up the leaves are on the left.
Continuity mistake: Mitsu picks up the arch that's lying against the tree, but it wasn't there when she knocked down Michelangelo.
Continuity mistake: When unconscious Michelangelo is lifted up and put on a cart, a man suddenly appears on the left.
Revealing mistake: When Mitsu knocks down Michelangelo, he falls on the floor and bounces, revealing the safety matt below.
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.
Other mistake: When the Turtles start their time travel, it's night in Japan. Almost 60 hours pass, and it's again nighttime in Japan.
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: 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.
Factual error: The movie feels the need to justify the fact that everyone in 1603 Japan happens to speak English (something most kid movies just gloss over) by explaining that they are trading with Great Britain, and therefore the locals picked up the language. That's real cute, but also blatantly false. Trade with the British empire was rather sparse, mostly with Dutch mediation, and knowledge of the English language was practically nonexistent - here, any peasant and even kids are practically bilingual.
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."