Revealing mistake: When Mitsu knocks down Michelangelo, he falls on the floor and bounces, revealing the safety matt below.
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)
Revealing mistake: It's even too obvious when the movie finale turns into a bit of a farce, but when Mitsu elbows the guards to free herself, it's obvious that she does no damage - the guards are wearing armor, even - and the extras don't do anything to 'sell' the hit. (01:20:20)
Revealing mistake: The rebels broke into the court of Lord Norinaga's palace. Donatello knocks down two guards with a leapfroggin' double flying kick. Which sounds way cooler than it looks, because it is obviously miles away from actually connecting. (01:15:15)
Revealing mistake: The grate on the prison floor (left open and unguarded after they escaped from there the first time) is made of thick metal...or so it looks, because when Michelangelo climbs up, it wobbles by the lightest tap of his elbow. (01:09:55)
Revealing mistake: Walker celebrates the news that he's gonna get the agreed price on his shipment from Lord Norinaga. He fires his cannon, which not only must be firing a cluster projectile since it disintegrates in a single shot 3 targets, but also it makes them implode. (00:58:20)
Revealing mistake: In a complete non-sequitur, the Turtles declare that they are friends, taking off their masks. Mitsu picks Raphael's helmet up herself; not only her hands chance place on the helmet at the cut, but the helmet is soft rubber under her fingers. (00:40:25)
Revealing mistake: When Raphael tosses the sai breaking Mitsu's arrow in two as she is aiming at April and Whit, before it cuts to the close-up the actress is smiling as she turns around, obviously pleased because the take of the throw worked. (00:39:45)
Revealing mistake: Walker meets his demise in sequence with a shockingly poor visual effect; he falls into the ocean from the cliff, and instead of splashing into the water, he sorta disappears quickly through it. (01:22:55)
Revealing mistake: The Turtles are preparing their time travel. Splinter tells them "The scepter will be in the temple, which means..." Pay close attention here, because now the rest of the sentence is obtained dubbing the line over a rewind of the previous footage. (00:21:50)
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."