I don't agree, because of the use of yellow color in damage bar, this could lead to misunderstading. Despite this, I agree with your proposal of making evident how many hp a deck has, but in a different way, the bar should be divided in 4 part:
yellow = damage you do in this turn
green = life remaining after your turn
black = life lost in previous turns, it can be recovered with heal or similar
grey = life lost due to deck age, it can be recovered only increasing your deck max hp (with SoD & SS)