I think the killer feature for these separate tools is usually that you can easily do a right click -> "copy as curl" in the network inspector, then import it in Postman/Paw and then tweak parameters / add headers there. This is not really possible in the browser network tools.
And then you accidentally refresh or close the tab and everything is gone. I usually use specific requests over many days if I'm reverse engineering something so having these available, sorted in folders for me is important. Of course for other uses cases it might be fine to have them live in the browser.
I just use Browser's Network tab for that nowadays. CORS can be a trouble at times, but that can be avoided with a few tweaks.