In general my response would be:
Hardness testing is performed to check the hardness (brittleness) of the material.
Is it required - varies by materials and codes, wall thickness and individual company standards and specifications
What is happening - If PWHT is not undertaken correctly in terms of heat up or cool down times or to the wrong temperature, significant changes in the steel material properties can occur
It is important to check the material qualiy after PWHT and a key indicator is the hardness. This can increase or decrease excessively and hence the material can become more brittle or softer, than the original, which could cause it to fail in service.
My motto: Learn something new every day
Also: There's usually a good reason why everyone does it that way