First of all, the majority of bugs go through PBE and are fixed on a later date. Some of them we have for years to come.
Many do. Many don't. The point is that they get found, and some of them get fixed. In particular, the ones that make the game completely unplayable or which make certain textures or particles completely disappear usually get picked up and fixed (you have no idea how many times Trundle's pillar has been invisible). I play on the PBE a lot, and bugs do get fixed in between patches. Even if many bugs don't get fixed, many others do get fixed because of the PBE. Other bugs get noted, and get fixed later, because fixing bugs can take a lot of time. The PBE has a job, and it does it. That is a better use for it than highly experimental testing.
And to expand, yes, you do have to explain yourself for every change, because this way you give the community a better understanding of your vision and the route you are taking to materialize that
That can be done without wasting resources on making the PBE do something that can be done more effectively in private testing. More communication on what champions they are and are not looking at would be a big improvement for Riot, but that can be done just as effectively without PBE changes. Straight communication also doesn't run the risk of having people get attached to a really unhealthy game change and getting pissed when it gets removed (and it would happen. If it buffed a weak champion, there are a lot people wouldn't care if it made every other champion suffer). There is a lot less risk in just communicating without presenting whatever highly experimental, possibly terrible idea a developer has to the easily-annoyed community. I mean, just a few days ago a Rioter said they were working on a tenative new Ashe passive that might come out later this season. I saw people swearing to uninstall the game if the changes didn't go through, and they don't even know anything about it. There is risk in showing people a lot of stuff..
I don't see much of that happening to begin with.
If it was happening, would you see it? It's private testing. You shouldn't be seeing it. You should be seeing good balance changes, and hearing responses to community complaints about champions. Seeing the full testing process isn't necessary for either of those, and it probably wouldn't be helpful.
Correct me if I'm wrong, but I think a lot of the basis for this idea is frustration with Riot's bad balance and lack of communication about certain champions, right? Well, I can agree with that, but I just don't think this idea is going to fix anything. It just seems like a way of forcing communication, but not actually using the PBE for any useful purpose. I think it would actually hinder testing a lot. I just want them to communicate more, not try to get people involved in the really experimental parts of design.