Characters in this story do not do the typical shoujo-style pushing/pulling people away from each others. Each has their own motivations based on their character and backstory, not just for the sake of the main couple/against them. There is likely no one "making them chase the same person so they can meet again". That... there's no reason for any character to play matchmaker here. Sensei has grounded every action in this story to character motivation thus far there's no reason for that to change.
In terms of Amou's motivation, my guess is that Amou feels kinship to Doumeki, understanding how it feels to have your world revolve around another person, feeling like they are your only purpose, and willing to live in the yakuza world no matter what because that is where your person belongs, he empathizes with Doumeki.
Amou feels a similar way (though not romantic/sexual, at least never stated to be) about Misumi, as Doumeki to Yashiro. Amou has been with Misumi for many, many years, and although can have his own thriving civilian career, decides to stay with Misumi because that is where he feels like he belongs. He understands Doumeki and knows that Doumeki may feel purposeless in life if he were forced out of the yakuza world, and thus away from Yashiro, forever.
All that said the one person that does do something quite shoujo-esque is Misumi, when he tells Yashiro to fire Doumeki the first time they meet because Doumeki is Yashiro's type (Misumi then reflects on it being like... woops that might've just made them want each other more).
Great explanation!
I think Nanahara asked for Amou’s help with placing Doumeki with that other group. Remember when Nanahara said he wanted to do something for Doumeki because he saved Yashihiro. Nanahara and Amou seem to have a connection, because remember when Yashiro said to not tell anyone about his eye, “not even Amou”. Whatever it is, i think it will be fully revealed later on, as Tsunakawa is looking for info about Doumeki’s past ties.
Why would amou-san, the right hand of misumi-san gives doumeki a hand? As far as saying he doesn't mind to blow the chief of sakura family just for him to take doumeki into his lot? What was amou-san purpose? He just barely know doumeki you know.
Maybe, just maybe amou doing this so doumeki able to takes yashiro away from misumi, because amou said don't tell about this to misumi even tho he is the right hand of his and favours him so much! And i think someone making them chasing the same person so they can meet again. Because yashiro made a weird face.