* Website sends HTTP response header containing a bitcoin address
* Browser keeps a bitcoin wallet
* Browser shows a button to send bitcoin from said wallet to said bitcoin address
And that's it.
You need a fee because otherwise nobody will do (or is doing) the work to implement it.
Of course, with something this small and basically one-off, given the number of open source devs building bitcoin projects I'd be surprised if no one's done something similar already.
Looking into this further, they would eventually email me after there are $10 worth of funds queued up for me, and then after 30 days of me ignoring them, those funds would be spread out amongst other sites who do want them.
However, until that wallet has at least $10 in it, Brave is just sitting on top of free money. I don't want that.
[edit] P.S, Brave doesn't identify it's self in the user agent header. I just downloaded the Linux version and it identified it's self thus:
Mozilla/5.0 (X11; Linux x86_64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/52.0.2743.116 Safari/537.36
And this means the users, who think they are contributing to you, actually aren't at all. This seems deceptive.
One reason could be they don't want to bug website owners about small amounts.