Akechi’s entire conflict is about his selfworth: He’s a product of a damaged system -the fostercare in Japan to be exact - and that makes him relatable to the rest of the cast since everyone comes from a shitty background in some way or another, but that’s where the similarities end.

From then on he’s defined by two things: 

1.- His background is very vague, having only his own words as only source, which are very unreliable because this guy lies through his teeth.

2.- While he come from a bad background like the rest of your party, his choices are extremely different.

The little we know is that he’s an orphan, his mother committed suicide and he knew she was resented because she had a son with a man while unmarried, Shido. When his mother dies he’s put in foster care until he’s old enough to live alone and during this entire period of time he’s frustrated, angry and terribly lonely

With no other direction, he focuses his issues in the only person he knows could do something about it: His father. 

What he really wants is to prove himself he can be worthy, because he’s a kid hungry for affection, but as you mentioned it has to come from his Dad. So he targets Shido to be useful to him and basically become the queen in Shido’s figurative chessboard.

The point is Shido is a corrupt man who has no reason to listen to him (he doesn’t know he’s Akechi’s father)  until Akechi kills the people from the first scandal around Shido.

Akechi orchestrated his meeting with Shido to a T and Shido doesn’t know why.

So trusting in him or trying to put him under his wing is basically shooting himself in the foot, so he gives him praise, but at a distance. Akechi sees this as a personal failure and settles for “killing him” instead, because only that will make Shido take him seriously.

And from there on, it has been 2 years and half. 

Akechi knows there’s people out there changing hearts so he doesn’t care for the Thieves until Madarame is taken down, because he’s the first direct Palace Ruler the Thieves affect Shido helped gain influence.

However he takes action AFTER Kaneshiro’s palace is down, the second ruler Shido had under his thumb and that sets his mind: There’s someone out there who’s out to get Palace Rulers, like Shido. 

Well, fuck.

If he wanted Shido dead he would simply guide you towards Shido’s palace, let you fight him and, while he’s in the back, shoot Shido in the face and cause a mental shutdown, like he does with Kunikazu Okumura.

When you face him as a boss he just tells you how much he wants Shido to recognize him as the son he is and cherish him for that while screaming at you how much he hates you because you’re achieving what he wanted, in his eyes, undeservingly. 

Cognitive!Akechi is Shido’s vision of Akechi and he gets Akechi perfectly, it gives Akechi the realization Shido always saw through his plans and knew what Akechi really wanted was his love and recognition because he suspects he is his Father.

So he knows in that moment it’s useless to continue his plan, his grand reveal to make him love him and cherish him, his plan of 2.5 years goes to the trash bin and JUST THEN he requests the thieves to change Shido.

Akechi, the character who’s bloodlust can be seen from the moon, who killed or hurt half of Tokyo’s population only to get to his Dad and who shoot you in he face without hesitation is asking you in his ‘last words’ just to change Shido’s heart, not to kill him.

The main problem is that his entire character is a web of lies that he either tells to others or to himself to be able to move forward: the denial defines his character and easily clouds his own feelings.