"Things like visions, spirits, and miraculous healings" seem (to me) insufficient cause to redefine the common, functional definitions of "universe" and "nature" to exclude things which exist. If I follow, you're defining "supernatural" as "existing things which are not subject to the laws of physics." Yet your awareness of your examples suggests they're subject to physical laws to the extent that people are presumably able to perceive them - Are our perceptions of presumably existing phenomena not subject to laws of physics? Is there a forgone premise that our conscious experiences are made of something finer than meat?
Though I appreciate your taking the time, I'm not seeing an argument more compelling than the god of gaps/Cosmic Squatch Fart. I don't know that "the physical universe cannot account for its own existence" means anything. What is the measure of accountability, in this case?
In certain cases, supernatural events and agents make an impact on the physical world. Does that make them part of nature? That's probably a question of definitions. What I'd emphasize is that, if things like that happen, something else from somewhere else is breaking what "should" be a closed chain of cause and effect. If things like that happen, I'd say in that sense they're not part of nature, even if they have an effect on it. The "supernatural" may have its own laws, but not OUR laws.
I think it's fairly self evident that consciousness is something non-physical. It may be exclusively generated by the brain (I'm not sure that it is, but just supposing), but that doesn't get you out of the problem. The experience of being conscious is something of a different quality, not just a different quantity. Now I'm not proposing something like a Cartesian soul. I have absolutely not the slightest idea what consciousness actually IS, and I've checked. But it's not physical, and it seemingly has causal power in the physical world, which is very strange. In that limited sense, I'd say it's not something entirely "natural". I'd go even further and say that you should be far more sure of your consciousness than you are of atoms and molecules, because all knowledge of those things come to you through your consciousness. If the model can't accommodate this, maybe it's time for a new model.
The universe can't account for its own existence means that the universe is a finite thing. That thing might not be God necessarily (although I happen to think it is). It might be something like the Dao. It might be some sort of previous state of existence where cause and effect have no meaning, where clocks run backwards, cats eat dogs, and Adam Sandler is funny. What it cannot be is that the universe as we know it just happens to exist. It may be that the physicists are right, and the big bang started from quantum fluctuations. Well, okay. Why those laws of physics, and not another? Why quantum "stuff", which is no less real than rocks or stars or chairs? Why anything? Why not nothing? The physical universe is finite, and finite things are dependent on other things. Either it's turtles all the way down, or the buck stops somewhere. If the buck does stop, it must stop at something that CAN account for its own existence. The only thing like that would be something that not only exists, but exists NECESSARILY. And that might be God, or the Dao, or some bizarre form of primordial potential. But it's not the universe as we know it.