Could you imagine Midori doing that?
Yes, actually.
Especially if that "someone" was Kana (or maybe Aoi or Naomi) and Amu finds herself on the hook for aiding and abetting in the murder of government officers, even if it wasn't true.
Even in the context this quest, where Amu has psionics and could otherwise force her version of the narrative into her mother's head psychically to show her it wasn't true, in the case that Amu is actually arrested and picked up by JPs first, they'd probably slap the psionic-equivalent of handcuffs onto her and stop her from using any psychic powers to "prove" her innocence that way (which is fair enough, I mean just imagine putting a mind controller in front of a court when they can use their powers).
JP's prosecutors would then drag into the open all of Amu's past fights with Easter and people like Lulu and Utau*. And then bring in Kana's mother to detail all the killings that Amu's friend had previously participated in, painting Amu as a deliquent who habitually gets into fights and abuses her supernatural powers and has now finally gone too far and joined a gang of killers. They would further bring up the fact that she has a split personality - 4 of them - and is therefore, additionally, mentally unstable and a danger to society.
By the end of it, Amu's mother would be completely dismayed and probably reasonably ready to believe her daughter was completely out of control and mentally ill and needed to be handed over to specialists with experience in psychic powers for treatment and who could teach her how to safely use and control them and not go around regularly mind controlling people on the streets with beams of psychic energy.
Her first reaction might be denial and disbelief, but realistically, this is going to be a trial that lasts at least good few weeks if not months, leaving plenty of time for her mother to look back upon the evidence presented and realized that - oh dear - Amu had, in fact, been getting into fights since her elementary school days. Amu herself would probably have limited opportunity to convince her mother otherwise, as she would probably be isolated in a JP's holding cell from the point of her arrest due to her psychic powers. Any visits from her mother would be behind a glass screen, likely flanked by guards.
Now, unlike what I assume the protagonist of P5 got, I figure that Amu would probably have a decent lawyer, as I assume that Tsukasa and possibly Easter would shell out the money for one as a favor to Amu herself. But even her crack lawyer isn't going to be to hide that fact that Amu having 4 Charas is not normal, even by Chara-user standards, and that she was definitely hanging around and being friendly with Kana and Naomi - even if they can successfully cover up the fact that Amu knew anything about them being killers, until whatever incident landed her in hot water.
In fact, Amu having a crack lawyer would probably be the only reason she even gets probation instead of jail time for helping a murderer.
But it wouldn't un-ring the bell that JP's prosecutors had rung in airing all her dirty laundry.
And if Tsukasa then said he knew a guy who lived in a coffee shop who could help to rehabilitate Amu and had experience treating psychologically unstable psychics, I can see Midori thinking it would be for the best if Amu went to him to serve out her probationary sentence.
Especially since there is another, younger child in her household who she wouldn't want turning out like Amu.
* I'm assuming the scenario is something like the "initially planned" storyline, where JP's was called in by Manticore to help hunt down the Scavengers and is not on good terms with Amu. Obviously, as the quest currently stands, Amu has Lulu (and by extension, Hotsuin) vouching for her and also her public demon-hunter reputation meaning JP's would be unlikely to go for a full prosecution off the bat, even if Manticore were urging them to do so.
