Ah, guilt and manipulation 101, call them out and all of a sudden it's tit for tat and withdrawal of support.
Don't engage in the back and forth, don't be tempted to say "no no i didn't mean that, don't do that" - that's the reaction she is after, so she basically gets a free pass to do what she wants, she calls the shots.
Instead, say something like
"you picking him up from childcare, I'm very grateful for, but is totally unrelated to this request to stop telling him off with his food, however if you feel like you want to stop collecting him, then I'll make other arrangements."
Your mum is a bit toxic.. start to recognise toxic, don't engage, don't start arguing, remain calm and don't give them the response they want. You're setting boundaries with your child. You are allowed to do that. Don't bow down to your mum just because she starts threatening you.