Maybe you missed my response that the rules do allow you to heal someone who is dead. Those heals, however, do not bring them back from the dead, because the rules say nothing about doing so.
So, then you're left with wondering what a dead character at full hp is like... and as far as I can tell by most definitions of 'dead' that just means they're still a corpse. And that's as far as the rules go. They decided not to, say, explicitly define that you couldn't act while dead because they assumed you'd figure it out.
Which brings us back to me pointing out that you can and should interpret the rules as needed, such as figuring out DCs to take actions not defined well by the rules, making rules to allow cool things, flexing DMG p42, changing the rules when you need for the fun and coolness of the table.
But that doesn't change a clearly written rule somehow. If you want a creature to be immune to prone it takes two whole words to make it happen. If you want a power to not push a creature, it takes a few more words but it's easy to do. Those rules exist for a reason and changing them on players by whim can hurt feelings and create a situation in which players are unsure of whether things will work, feel unfairly put upon, or just don't bother trying things. That's not good. Figure out changes you want in advance and decide them, don't just make changes on the fly. Pressing Strike can't push because the halfling is too small, but the Invoker's slide 1 at-will is fine? Why - because it makes more sense to the DM? What, the halfling can't stab the giant in the foot and make it stumble back and the invoker's sun strike is equally as powerful at moving tiny creatures as gargantuan? Make rule changes to make the game more fun, or make them in advance so your players know.
Now, I'll concede that the bloodied argument is slightly less cut and dry than the whole prone an ooze or push a primordial argument. But only slightly, and I do wonder how much is wishful thinking because of Consecrated Ground. Thankfully there are very few powers that are impacted - things like powers (or a dragonborn paragon path) that work only while you're bloodied and turn off when you're no longer bloodied for example.
Now, healing the dead might not be intended to work, but frankly it's immaterial. You could make an argument that it doesn't work because a corpse is an object, and no longer a creature, which removes it from target selection... but why would you care? Healing the dead has literally no effect on the game.