In the latest major release (384.4 and beyond), in Chrome and Firefox, if I click on the "Browse" button to choose a firmware to install, the browser closes.
Correct. The behaviour you describe isn't at the HTML/Javascript level, but at the browser itself's level (Javascript can't close your whole browser). Beside, I do all my test and development using Chrome, never experienced this in the dozens of flashes I've done with that version.