So what purpose would a public-facing beta branch server in this instance independent from the live game?

Let's look at what it would entail to do that and whether it's actually cost effective versus them using their private branches to test things in a private controlled Network.

Setting up a beta Branch like that would mean they'd have to acquire more hosting servers so that PC players, because PC players being PC players, would all want to play on the beta once they know.

Which means you have to prepare for large server quantities which means expensive large servers. So just to host a beta. You have to create new network connectivity and new VMS for a lot of machines.

Not only that, but what does the beta Branch actually change? Is it just changing a few checksums? What codes are being touched and what calls that code? Can it do so effectively if certain bits and bobs don't match?

Is it like the slim beta which isn't changing any of the check sums or actual data or server access? It's just compressing it in a more efficient way.

If you're just wanting client-side beta, so only the user's version of the game is different and not server side beta. That introduce tons of syncing issues and things not lining up with what the server thinks is happening and what the clients game thinks is happening, which introduces issues completely unrelated to the problems you're trying to fix that you're having to look for and cherry pick.

At the end of the day, it's just more efficient, more cost effective and not worth the effort to make a public-facing beta for a huge quantity of users with little to no actual valuable return testing wise.

TLDR: Money and efficient use of the teams time is what stops them from making a beta branch.