It kind of gets down to the core of magical power in the HP universe. One of the lessons from the first few chapters SS is that the “stuff” of magic is unnecessary. Remember Harry makes his hair go back the way he likes it and makes the window disappear without even knowing what a wizard is, no wand, no training, just will power and wishful thinking. Wizards in the HP universe seem to be able to just make stuff go, sure they mostly use their wands for big things but the wands aren’t totally necessary, words are good for focus but they’re even trained in how to cast silently, plenty of “throw away” magic is done throughout the series with no wands, words or even gestures of any kind. All that stuff is a crutch to help the wizard focus. Even non-throw away stuff now that I think a little more, notice there’s no wandwork or words to go with animagus stuff, they just change, that’s probably why it’s considered so hard, it’s big magic to start with but it’s completely crutchless.
And then there’s the psych out factor, a lot of the training the kids seem to go through, especially the Dumbledore’s Army training but other stuff to, is largely to believe that they can do it. Most of the setbacks they have early in any training come from lack of belief.
Which brings it all back to the Hallows, especially the Elder Wand. I think most of the power of the Elder Wand comes from the psych out, and the personal belief. I think the wielder of the Elder Wand gets a boost of confidence, and people facing them get psyched out. This would also explain why the “rules” of the Elder Wand are so inconsistent, if the wielder is over confident and their opponent refuses to be psyched out then the “unbeatable wand” becomes pretty beatable. It’s probably nothing more than a really good wand (whatever that means), maybe with a few standing spells on it that make the holder more resistant to certain things. But given that magic is so belief and willpower driven a wand with a kickass rep is worth a lot, even if there’s nothing at all special about it.
I've been thinking about magic in HPworld, the same way I think about woodworking.
This wand discussion would be be something like the difference between good and poor woodworking tools. The tools do make a difference in the finished product -- good ones do a lot to enhance precision and smoothness of line. But just having good tools doesn't really mean much if the person using them is an idiot.
The real "magic" of woodworking still depends on the person using the tools. Patience, ability, training, practice, and a sense of the art of it, are among the many personal factors that make the difference between a hacked up bunch of boards, and a really nice piece of furniture.