As you and b1daly point out the majority of the cost of running Bitcoin is currently paid for by mining rewards, but even so the fees are still crazy!
I don't know what will happen when the mining reward disappears. There will surely be far fewer miners, which will decrease the real-world cost of mining a block (and also decrease the security of the system!)
If proof-of-stake or something like it is ever adopted then presumably the cost of running the system will plummet but I'm not familiar enough with that stuff to say much more.