For example, if we go to the OMDb API example page, search movie by title by entering "test", we get the following response:
Now if we open up Chrome devtools, go to the network tab, and hit Search again, here's what that response looks like:

Is there a way to filter network requests in ANY browser by a specific value in a payload's key value pair? When I type "united states" in the filter field, I expect a request to show up because there is a payload that has that string:

But it doesn't seem to work in Chrome, FF, or Safari. Anyone know of another browser that does this? Or maybe an extension or other method of achieving this?

Have you tried using search? That's the way to filter payload. The filter textbox only filter header information mostly, not searching.