When your WH is on the lowest floor/basement, I would not put in a vacuum relief unless the local laws called for it. For a WH in an attic or upper floor, then it is called for.
The same considerations apply to a water softener. I don't have a vacuum relief for the water softener either.
But as a way to make the WH drain faster, I would not bother. If opening the tub hot does not provide fast enough flow, then I expect the valve or debris at the valve to be the limiting factor. I would try blowing out the debris by turning on the water and letting the water pressure push. But I still think opening the T&P temporarily is fine to admit air.
The same considerations apply to a water softener. I don't have a vacuum relief for the water softener either.
But as a way to make the WH drain faster, I would not bother. If opening the tub hot does not provide fast enough flow, then I expect the valve or debris at the valve to be the limiting factor. I would try blowing out the debris by turning on the water and letting the water pressure push. But I still think opening the T&P temporarily is fine to admit air.