*Your PFB will be a toddler in no time and will be boisterous at times, throw toys, be less than gentle around younger children, hit another child, snatch things, and maybe bite.
My child is out of the toddler phase and I haven’t seen any of this behaviour at all. It’s really not inevitable if you teach them how to behave.*
Here's the thing, though, all children are different. My DD1 (now 10) has serious anger issues because of being adopted and SEN. She can get violent, throwing things, and has been violent to me at various times. There's none of that with DD2 (7), despite the fact that they're full birth sisters.
Your toddler didn't do those things but if you were to have a second child you might have to eat a good dose of humble pie.
Having said that, the mum should certainly have been supervising. Although lecturing her toddler was pointless, he was hardly going to understand was he? Just a firm 'no' was required and gentle redirection.