Default profile or default browser? I have 2 profiles and it got both of them ok. However, I got the Not-Default-Browser message so registered it.
How are you starting it? For 16.04 I changed the /usr/bin/firefox link to /opt/firefox/firefox (originally it points to some script).
I saw a lot of solutions including the home directory but that was the first one I soundly rejected.
EDIT: Per your edit, yep, that’ll run as root. The method I used works on every method of starting firefox I can find, including opening .url files.