You're missing the point; it's like me saying she's an ex for a reason - reasons we don't know and would never get a straightforward, honest reply. Our perception of our wrongs or someone else's varies greatly.
She also stated that this is something that has not happened before, the pushing, so his flaws could be leaving the toilet seat up, for all we know, she could have cheated on him, and they have split up.
He sounds perfectly well to parent children; there is nothing wrong with wanting your children to be respectful and obedient; after all, they are kids, are you suggesting that he lets his children do what the hell they like and treat him like crap? I'm sure you're not, because that would be bad parenting, and your name suggests you believe you're a good parent. I believe you're a good parent, but I also believe I wouldn't agree with all your parenting styles and decisions over the years, and that's fine, as I said, anyone who thinks they have no faults and are the world's best mom or dad is just laughable.
We still don't actually know what happened in that split second; we're getting a redacted version from the one who believes they have been wronged. For Dad to put his son outside and call Mom to collect, I'd say it's pretty serious in his eyes, and this is where our perception comes into play - he sees it as a big deal, while I might brush it off if it were my kids.
The OP is doing what she believes is right, but it also sounds like she's not entirely sure, as she's here asking for opinions or validation on whether what she's doing or has done is the correct way to approach the situation.
For me, she has to play the piggy in the middle, even if she doesn't like it. They are no longer a family unit, hence why this has dragged on for six weeks, as I'm sure if they all lived together, this would have been done and dusted on the same day.
In contrast to this story, my brother went through a bad separation. His daughter physically attacked him to the point that the police were called, and after the lies his daughter told the police, he was arrested. It was only when his son stepped in and told the police what had happened. He'd filmed some of it, and they released him, asking if he (my brother) wanted to press charges against his daughter. He declined, and he still had to leave his house that same day. If his son hadn't been there, he'd have probably been done for assault.