The abilities are still there, but have been changed to simple cooldown (1:30 min). So, technically you don't need to melee to use them. It's still benefical IMO, since even if healer melee damage isn't great, it's still helping to kill the opponent.
Sitting down during combat gives rested state, but the regeneration is much lower than outside combat. This can be neccessary at times, but as you said, in most situations it's better to fight, and rest after combat ends.
Ah ok, yeah I remember them being like 30 seconds. Still, even at 1:30, I think you are probably better off building more readiness for the next combat at least until the bar is full and then sitting. Or as you said, doing even marginal dps helps more than the negligible mana you are going to regen while sitting during combat.
Perhaps this will change, or scales differently at higher levels. But at least at lower levels the mana regen during combat simply isn't worth it IMHO.
It's dependent on your class, what level of gear your tank has, the competency of the group, yadda yadda, but yes, sitting in combat sometimes is absolutely a requirement, sometimes, it's not - simple as that.
Edit for:
(As a Cleric, solo healing)
Level 4 Tank, brand new, fighting Thugs in Cloth = Sitting a lot. Like mostly.
Level 4 Twink Tank wearing some chain fighting same mobs = You decide.
Don't be blindsided thinking all things are equal ":)
-But saying 'sitting doesn't matter' is kind of like thinking "Inches don't matter" in a Football Game.