Continuity mistake: When Benkei climbs the sewer ladder and disappears into the light, Donatello is touching the ladder in the first shot, in fact he puts both hands on deliberately, but he's not even touching it in the second. (01:28:35)
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.
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)
Factual error: During the movie, characters may use lingo that would not be understood easily in the time period of their counterparts. April for instance uses the word "cool" in the modern sense, and nobody questions that, but you could assume that they don't understand every word and infer nonetheless the meaning from the context. Walker however when he's about to fire his cannon uses an idiom "Try this on for size" that was coined in early XX century. (01:19:40)
Continuity mistake: In the baffling phase when Whit takes exception to the killing if it's done by guns and Walker just throws him with the others, Elias Koteas moves his gun from the left to the right hand when Stuart Wilson calls for Niles, but in the next shot the gun is much lower. (01:18:15)
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)
Factual error: Raphael tells Yoshi that being a kid, he shouldn't think about war but rather fly "one of those things." He is pointing at kites who have artwork (kabuki-ish makeup with the modern japanese flag) that looks out of place for 1603. (00:56:40)
Other mistake: Apparently, Whit is the only prisoner in Lord Norinaga's dungeon who wants to be set free; everyone else does not react to the fight, ask to be released or struggle in any way. (00:33:00)
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)
Continuity mistake: When the guard tells Norinaga about his honor guards; "They're coming! They're behind the wagon!", Norinaga's hand placement changes between shot and reverse. (00:28:15)
Other mistake: Not finding his scepter anymore, Norinaga speaks in English to the guards. It's rather stupid of him to do so since it's counterproductive to show vulnerability to his 'ally' he wanted dead at the beginning, but let's forget that. The problem is that his envoy addresses the Turtles outside the gate in English as well. He thinks they are the honor guards, fellow countrymen, there's no reason why he'd use English. (00:28:00)
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.
Character mistake: Donatello is running a rather verbose program analyzing the scepter. On screen the text mentions the "time-space CONTINIUM" rather than "continuum." (00:17:45)
Character mistake: Pretending to be a witch, April says that the song the walkman played was obtained because "I shrunk four incredibly cool guys and I locked them in there." The Barrio Boyzz were five at the song's release. Did she let one Barrio boy go? Was one of the Barrio Boyzz uncool? (00:16:40)
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."