Questions and thoughts like this are amusing. Someone appends the word "neuroscience" to a question or remark that is, in a way, about neuroscience. But it is also capable of being thought of in simpler, broader terms.
What I'm trying to say is that, it seems to me that, more often than we might expect, neuroscience is about… other things. (Unless you work in a lab somewhere.)
For example, take the question "How can I use the power of neuroscience to forget bad memories?" Practical, useful answers probably have very little to do with neuroscience itself.
I believe it’s both more helpful and accurate to think of “forgetting” as one's brain recontextualizing and reshaping itself.
I’m also not convinced that “forgetting” is a completely achievable objective in the first place--to literally be able to erase a memory. Episodic and long-term memories tend to stick around. This is a feature of the mind, not a bug.
For example, our brain remembers that time we burned our hand on the stove, so in the future, we don't do that again. Initially, we wince. But later, our brain molds itself around this memory. Without the ability to store information over large time scales, neither language, relationships, nor our own personal identities would develop.
Your memories probably don't change much. It’s your perception that rotates as you learn new things about your memories. There’s a lot of literature to support this hypothesis (brain plasticity, constructivism, memory consolidation, etc.).
But our brains are also imperfect—prone to distortion, illusions, and biases. Sometimes it wishes it could more clearly remember things. And other times, it wishes it could forget them.
In the spirit of Hebbian theory—"what fires together, wires together"—it seems to me that the only way to even come close to “forgetting” a memory is to replace it with a more powerful one.
The primary paradox of the past is this: the past has a grip on us, not because of the past itself, but because we're unable to conceptualize things that haven't occurred.
In the end, I tend to believe that forgetting is more about changing how we understand and relate to our memories than anything to do with forgetting itself.