What problems arise in responding to Gettier problems with an
assertion "the formal definition of knowledge, as justified true
belief, does not need to exactly correspond to intuitive notions of
Gettier problems are that some ideas that allegedly satisfy the formula of being justified true belief are not knowledge. One example is suppose that Peter believes that he has ten coins in his pocket, that he believes he has ten coins in his pocket and that he is going to be given a promotion to chief toilet cleaner and has heard about from a reliable source. The Gettier claims Peter has a justified true belief that the man being promoted to chief toilet cleaner has ten coins in his pocket.
The Gettier problem highlights a real problem in an indirect way hidden by a lot of bad ideas. The problem is this. There are many facts that nobody gives a hoot about and don't seem particularly significant, like the colour of my computer mouse. Knowing the colour of my mouse won't shed any light on any issue. So people don't think the information about the colour of my mouse is knowledge, and they are right.
Your response is that this is just a problem of people having a false intuition, but you are wrong. This is an instance of a fatal problem with the JTB theory. A theory is deemed to be justified if it has gone through some process called justification that make it true or probably true. The model for how JTB knowledge is created goes something like this. You somehow (1) come to believe a theory and it is somehow (2) justified and (3) it is true.
JTB theories typically either have nothing to say about step (1), or they say something that is false. For example, inductivism (the belief in induction) claims that you somehow get a theory from experimental results. In reality, there is an infinity of ideas compatible with any experimental result, so you can't get a theory from experimental results. So then where do theories come from? Where could they come from? They can't come from the truth since you don't know what's true and they're not totally random. What happens is that you notice a problem with existing knowledge, something existing knowledge doesn't explain, and then you produce variants of that knowledge to try to solve the problem. You look for criticisms of the variants, including looking for cases in which they are incompatible with experimental results. You keep coming up with variants and criticising them until only one is left and the last variant standing solves the problem. You then look for problems with your new theory.
The growth of knowledge starts with problems. It starts with current, flawed knowledge. The knowledge it invents solves those problems. Knowledge is always a solution to a problem. The ten coins in the pocket example is just an example of somebody happening to be aware of some useless fact. The JTB theory doesn't explain why this is not knowledge. It also fails to solve a lot of other problems. JTB is false.
For more on why JTB is piffle and what should replace it, see "Realism and the Aim of Science" by Karl Popper and "The Beginning of Infinity" by David Deutsch.