You make it sound like they politely asked if she would please change seats, with the decision ultimately up to her. But Coulter says she was ordered to change seats, with the implication that she would be removed if she refused.
Coulter suggests the only explanation provided was some sort of undefined “emergency”. The woman who allegedly got Coulter’s seat doesn’t appear to be disabled. The comfort and convenience of sitting with family (if that’s all that it is) hardly constitutes an emergency.
And it may be the case that Delta reserves the right to reassign seats without explanation, but even if a refund is offered, that hardly makes it all okay. To behave in such a manner, to just (it seems) arbitrarily remove someone from a seat they paid for defeats the purpose of offering pre-booked, specifically selected seats. Why offer the service/product if you’ll just cavalierly jettison it?
I get that you don’t like Coulter, but until video surfaces showing her account to be false, or Delta offers a just reason for taking what she paid for from her, then Coulter is in the right.