The Firejail profile for Firefox-based browsers allows the application to interact with the ~/Downloads directory by default, but I took a look and it appears the profile for Chromium-based browsers does not. You would need to add it as a whitelisted directory in a local Firejail config.
~/.config/firejail/chromium-common.local
whitelist ${DOWNLOADS}
I agree that this is a case which seems overly restrictive because it breaks a fundamental feature of the application. I’m not sure if that is an oversight, or if there is some reason they set the profile up like that on purpose.