It does not have any inherent value, I do not need gold in my everyday life and I do not know anybody who does.
Having value is just a way of saying people want it. I want it, I bet you wouldn't decline if I offered you some at $4/oz.
And regarding inherent, would you rather have something with a little inherent value or more 'non-inherent' value whatever that is. It's a non-distinction.
...there is no way for you to know whether there actually is in the safe the amount of gold the bank says there is, this is what led to the fractional reserve in a first place.
Banks that can't convince people of their solvency will just go out of business in a free banking world. Maybe there is no 100% fool proof way to know the bank isn't tricking you, but it's the same with hamburgers. You can't be 100% sure they didn't water it down with dog brain, but my burgers taste pretty damn good, and there are things you can do or insist that they do to make it better. Watchdog groups, audits, glass safes, , etc.
We don't really have these technologies because we have a law instead, but that doesn't mean it's impossible to design ways of proving and verifying things.