Why can't I help feeling that if Mobius had shown Loki this memories he would have said Odin had died because Loki had banished him. Same about Thor. Your brother is dead and you too are dead because of Mad Titan. These memories were very well manipulated. Loki doesn't know Odin would have sentenced him to death. He doesn't know that he would stay in a cell for a year. That Frigga defend Odin's bad parenting. That Frigga died because of Aether, not Loki's words. That Thor is alive!
I completely agree with you. That is what Mobius would have done. What he did was psychological manipulation and gaslighting. He lied and explained the events in a way to break Loki's walls and make him vulnerable to accept his POV and offer. Which is actually a method of torture in interrogation that is specially used on political prisoners where I live. The part when Mobius lied that Loki was responsible for his mother's death specifically reminded me of them and made me sick to my stomach. What they do is that during many physical and psychological torture sessions they tell the prisoner who is been kept in isolation from the outside world for so long(in solitary confinement with no contact to their family), that one of their parents(usually their mother) has died as a result of their political acts of resistance and fights(because they couldn't bear so much worry and anxiety for their child in prison). This usually results in the prisoner breaking and accepting whatever they want from them and in some cases it cause them to commit suicide.
Now imagine what kind of psychological torture Loki went through by what Mobius said to him, right after his time with Thanos and his history of suicide. Which is why I'm baffled that many fans and also the creators consider it as therapeutic and what Loki "needed"! I don't expect that people be aware of this form of torture. But I thought the difference between psychological manipulation and a therapy session is obvious. Apparently not. I guess they think Loki needed that because they see him a villain who deserve this.
And don't get me start on thinking that Mobius cares about Loki, or according to creators he is inherently good! You don't constantly belittle, manipulate and insult the person you care about. Mobius is an agent specializing in catching dangerous variants. So the way he sees Loki is a tool to help to him catch this variant who is escaping TVA and for doing that he uses every way he knows like psychological manipulation and using Loki's weakness against him.
The idea that Mobius is inherently good is laughable. He is working for a fascistic/totalitarian organization that inflict its will on people, killing them(according to Miss Minutes introduction for sth as unimportant as being late for work!), doing acts of slaughter an genocide because someone dared to break the rules of their "sacred timeline"! Rules that no one knows exist in the first place! And he is OK with all of that and has the audacity to condemn Loki for his actions under mind influence and torture. And he doesn't question his ideology. Even when Loki challenges him to do so, because it gives him his glorious purpose. He is the example of every person who works in corrupted systems because it gives them some privilege and they're so lazy and inconsiderate in thinking, that they don't care how it affects the people who are suffering or being oppressed. Sure, he shows care and emotions towards some people, but selected morality is not inherent goodness. To TVA and Mobius the law breakers are not even people, they're "variants" and they don't treat the variants like beings who have right to exist and choose their own path.
In fact, Mobius and on a larger scale some of the TVA agents are the exact examples of the concept of banality of evil . That evil acts are not necessarily done by evil people. Instead, they can simply be the result of bureaucrats dutifully obeying orders. People who don't consider the consequences of their actions and don't question the propaganda and ideology they have been brainwashed with.