It’s definitely the responsibility of the seller to ensure an item arrives in one piece,
In the same way that a seller has to refund/replace an item lost in the post, the same applies to a damaged item.
Offering a choice of couriers is good, but ultimately it is still your responsibility as a seller to make right a damaged order, and certainly NOT up to the buyer to either take responsibility or put in a claim for damage themselves.
Responsibility certainly doesn’t end at packaging correctly.
Also - would the customer have to arrange pickup/couriers themselves?
I think that could be a bit of a minefield organisation wise, as many couriers won’t specify a time, and others like an item dropped off.