The problem is that if your definition of knowledge requires only three conditions (belief, truth and justification), then it has counterexamples, namely, the Gettier cases. So we need to either revise our concept of justification (for example, make it so that a knowledge-apt belief is justified only if it is for the right reasons -- but then, how to account for justified but false beliefs?), look for ways to supplement the analysis with additional conditions (for example, that the conditions are satisfied in a robust ways counterfactually), or reject the idea that knowledge can be analyzed altogether.
To make it more concrete how this sort of thing can get complicated quickly, consider your own explanation of the case. As I take it, you think that the issue is that "the justification is faulty". But how is it faulty, besides not being true (remember, we have reasons to want justification to not imply truth)? And also, there are Gettier-like cases where we cannot say that justification is faulty or where we can't find false premises (for example, cases where it seems like the problem is that the individuals in questions are not in an appropriate environment, see [here](https://en.wikipedia.org/wiki/Gettier_problem#False_premises...)).
Thanks for engaging in the conversation. When I say "the justification is faulty" I just mean it's not deductively air-tight, therefore we can't assume that it leads to truth. In this sense, pretty much all justifications are 'faulty.'
The thing I'm struggling with here, ultimately, is truth. All of these problems rely on some outside, oracular knowledge of what is true, (e.g. in the example you linked there really is a cow/sheep in the field, or Mark is really hiding under the desk). But we have no such oracle to refer to, so trying to ground knowledge in truth seems like a lost cause in the first place. All we can have is more or less certain beliefs.
Sure, from a present perspective on epistemology this may make sense, since we now know all the troubles and faulty attempts to decide proper justifications from incorrect ones after Gettier's paper. However, back then the idea to define knowledge using JTB was on the table. The knowledge as justified true belief view is about a definition of knowledge, not a learning method or how to gain knowledge. Many authors would even have rejected the idea of graded belief entirely and said you either believe something or don't (many still do).
Since you accept the notion of truth (otherwise you couldn't be a Bayesian), you'd have to explain why you reject the realist conception of knowledge inherent to the JTB view. Gettier's paper is one attack on it, arguing for graded belief representations is another type of attack on it. In this context it is worth noting that graded belief and categorical belief are very hard to reconcile because they have different logical properties [1]. It's known as Locke's Thesis and quite a vexing problem.
That being said, I share your intuitions. The factivity of knowledge has created more problems in epistemology than the notion of knowledge was supposed to solve. The German word Erkenntnis in Erkenntnistheorie has a meaning closer to learning theory. AFAIK there is no good equivalent to this in English.
To make it more concrete how this sort of thing can get complicated quickly, consider your own explanation of the case. As I take it, you think that the issue is that "the justification is faulty". But how is it faulty, besides not being true (remember, we have reasons to want justification to not imply truth)? And also, there are Gettier-like cases where we cannot say that justification is faulty or where we can't find false premises (for example, cases where it seems like the problem is that the individuals in questions are not in an appropriate environment, see [here](https://en.wikipedia.org/wiki/Gettier_problem#False_premises...)).