Depends on the situation, the person, the dog.
My old dog died on the 27th of Dec, I worked the 28th as scheduled, I needed to keep busy and whilst my job is talking to people about their dogs, I am ok with that. His death had been expected for some time - some considerable time in fact as he was just turned 17. It was not a traumatic death either, he woke up, vomited blood, we knew that wasn't something that could be ignored nor would we investigate so he was put to sleep later that morning.
However in the past we have had traumatic deaths both of young and old dogs which have hit far far harder.
I've had lots of dogs and therefore lost a fair number of dogs - two of them left me absolutely in bits for weeks, barely able to trundle on and function never mind work. One of those was a sudden traumatic death, the other was not and I was shocked at my own response, but no more able to control it for that.
We can't know in advance how we will feel about events in our lives - some people will be more than capable of going to work and trundling on and being useful and productive... and some people just won't be. For those, they're better off staying at home than trying to work.
Whether thats AL, unpaid leave, paid leave - thats between them and the employer. Maybe if you have a sympathetic employer, you'll be grateful for that compassion at some point in the future. Or perhaps you'll be lucky enough never to need it.