"I know I fathered Alston" .. the reasons it may be true or false are indeed independent of brain state. But "knowing" is not about whether it is true or false, otherwise this whole question becomes tautological.
If someone is just going to say "It is not possible to know false things", then sure, by that definition of "know" any brain state that involves a justified belief in a thing that is false is not "knowing".
But I consider that a more or less useless definition of "knowing" in context of both Gettier and TFA.
I wasn't talking about whether it was true or false that I know I fathered Alston. I didn't say anything about knowing I fathered Alston at all. I was talking about whether it was true or false that I fathered Alston, which (I hope you'll agree) is not a question of my brain state; it's a question of Alston's genetic constitution, and my brain state is entirely irrelevant.
I think that, without using a definition of "knowing" that fits the description of definitions you are declaring useless, you won't be able to make any sense of either Gettier or TFA. So, however useful or useless you may find it in other contexts, in the context of trying to understand the debate, it's a very useful family of definitions of "knowing"; it's entirely necessary to your success in that endeavor.
How about "beliefs that seem to be true are not necessarily true, and the causes of those beliefs may not be valid, especially if examined more closely"?
Or, try renaming the variables and see if it still bothers you identically.
If someone is just going to say "It is not possible to know false things", then sure, by that definition of "know" any brain state that involves a justified belief in a thing that is false is not "knowing".
But I consider that a more or less useless definition of "knowing" in context of both Gettier and TFA.