This is an interesting take - is my "attention" really worth several thousand a year? In that my purchasing decisions being influenced by advertising to that degree that someone is literally paying someone else for my attention ...

I wonder if instead, could I sell my "attention" instead of others profitting of it?

Yes, but your attention rapidly loses value the more that your subsequent behavior misaligns with the buyer’s desires. In other words, the ability to target unsuspecting, idle minds far exceeds the value of a willing and conscious attention seller.