Ok...so Archie's mom, she doesn't come off as abusive, but she does come off as stupid, and unable to understand her child, the picture of him flexing with her doesn't seem forced, it seemed like a bonding moment between a woman who probably greatly cared about her appearance before having a kid and kept up with fitness (considering ring girl and stripper), and a child with ADHD and unbridled energy, so she probably did encourage him to do sport/gymnastics ect because that was something she could understand and connect with him on, but other then that I think Archie spent a lot of time by himself, sad, and this is where the stupid Mom part comes in, she yelled at him a lot due to not understanding why he wasn't "normal". She probably had a go at him any time he was loud, didn't listen, wouldn't stop moving....so when the rabbit prank happened she probably again had a go at him, told him to stop being a little shit and fuck off.
And that was enough. 12 year old boy probably just starting puberty or at least something going on hormonal, mixed with ADHD, and most likely depression.
Pl-ish? as a young girl of 12-13 when my parents would yell at me or I'd get in trouble the during the teenage screaming match Id yell "Youd be happier if I was dead" but being rational, and not depressed/ADHD/tism or anything else it was dramatic but empty threats/words ...but with him he probably thought that and in a spur of the moment thing necked himself, thinking in fact she would be happier with him gone.
At least thats what I want to believe and this is her feeling guilty for all the yelling and being hit with the realization her baby is gone. Grief is a hell of a drug. Some people curl into themselves, others lash out, kick fight and scream and thats what I think we are seeing. She is trying anything to get him back to probably be able to apologize for being a shit mother
(Edit for spelling and rewording)