I've used Chrome too, same problem.
I am looking for something that saves just the JSON responses from a website. I can see the JSON in devtools network tab but can't extract it. I'm not a webdev so it may be easy I just don't know how it's done.
It's all there. The problem is deciding how to add code to save it in a rational reloadable manner.
Compared to eg ZFS snapshot: low cost copy on write, saved state. (Admittedly of data on disk, which is the goal here, but then the browser cache is on disk)