I think in a5e, the death save the hit die spending or even being hit by healing word will give you a le sl of strife & fatigue when whichever brings you back.Well, in o5e when you're at 0 hp you're unconscious and incapacitated. Thus you cannot take any action or reaction. Spending HD is not classified as either, but at the same time it's not a regeneration like effect. I don't know if any of this changed in a5e.
Sure, but because the death save may result in dying while spending the HD will always get you back to consciousness and positive HP, I think it's pretty much guaranteed there will always be one choice, i.e. spending HD.