Other mistake: After Leonardo revives Yoshi, he picks him up, hugs him and touches heads with each other as they hug but Yoshi pretty much moved his head immediately and Leonardo continued sitting there as if their heads were still touching. It looks like the actor didn't realise the child moved away and they never reshot it. (00:48:15)
Other mistake: When April O'Neil arrives bringing gifts, she leaves the manhole cover completely open - there's also an awful lot of light coming from there, as if the abandoned subway station were built inches away from the surface. (00:05:20)
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.
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)
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)
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."