No employer is expected to pay a member of staff for time off when a pet dies or in the other circumstances you mention. Often people take sick leave, unpaid, flexitime or make time up in those situations.
Many employers have realised though that helping employees in times of grief, loss, heartache are the things that build loyalty, flexibility, commitment and encourage hard work.
My employer was wonderful to me when I lost my parents, friend and dog. I never forgot that, it endeared me to the company and my manager and made me really want to do my best for them in the future - which I did until I retired.
When I ran my own business I was really kind and generous to my employees and they were great to me too. It came from my heart but it was incidentally quid pro quo too.
An employer who expects a stiff upper lip, an emotionless response when you are on your knees is likely to be shown the finger (metaphorically or otherwise) when extra effort or input is asked of an employee in future.
This kindness and decency produces the same result whether the employee is a clerk or a senior manager.
The posters on here who think management should have no life/emotions obviously think that when you become a manager you cease to be human.