It is definitely possible although rather difficult for even a very good technician if they don’t have the right equipment. But there is absolutely no need to change the whole body unless it is too damaged to put an unbroken screen or glass on without risking damage to the new one but I think your phones body looks almost perfect.
I, personally, would prefer to just change the whole LCD assembly but that will certainly drive up the cost. I do not own ALL of the equipment to replace the glass alone but I have done it at the customer request multiple times and usually don’t charge more that like $100-$130 USD but because I don’t have all the equipment there’s a chance the LCD could be damaged in the process and I warn them ahead of time. If they are ok with the risk and chance of damaging the LCD I give it a shot.