Firefox 71.0 is not working on Ubuntu 16.04 LTS

Looks like you've added a PPA that's meant for the team that's working to integrate Firefox, Thunderbird etc. into Ubuntu

71.0~b6+build1-0ubuntu0.16.04.1 500
    500 http://ppa.launchpad.net/mozillateam/firefox-next/ubuntu xenial/main amd64 Packages

That PPA seems to contain preliminary, "work in progress" versions. You might want to remove the PPA from your list.

Command to remove this PPA:

sudo add-apt-repository --remove ppa:mozillateam/firefox-next

To force the older Firefox to use your profile, invoke it with:

$ firefox --allow-downgrade

You might also want -P in there, to pick which profile is used.

(Source: Mozilla Support)

I had the same experience as you with the Firefox Beta PPA: after upgrading to 71.0b8, it refuses to run. I then downloaded Firefox Beta directly from Mozilla. That was also 71.0b8, but it still claimed my profile was from a newer version, and required --allow-downgrade to use it.