Continuity mistake: Whit and April are fighting by the spring, and just before it cuts to Mitsu preparing an arrow, Elias Koteas is pounced by a guy wearing colorful rags in black, orange and yellow patchwork, a blue and white pattern bandana and black and white striped legwarmers. Behind Mitsu, meanwhile...a guy with the same horrid fashion taste is charging. What are the odds. (00:39:35)
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 ninja-Michelangelo is on the horse, the sceptre swaps hands between shots. (00:25:05)
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: The pattern of the fabric below the sceptre changes between the first and second shot. (00:09:41 - 00:10:07)
Continuity mistake: When the scroll falls downwards, it is smooth, but from the immediate next angle it's got a big wrinkle. (00:09:54)
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.
Continuity mistake: When the turtles jump over the table, the tablecloth suddenly becomes way shorter between shots. (00:04:01)
Continuity mistake: For his curtain call, Stuart Wilson acts in close-up like he's losing balance, body arched back. The impact looks all different in the wider angle. (01:22:50)
Revealing mistake: The Turtles give chase to Walker; Leo and Raph run across a roof, but the tiles are obviously fake; look at Raph's feet and you'll see one in particular bending like gum under the push. (01:21:00)
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."