Would board of E-4/E-6 soldiers and field grade officers, who are focused on problem-solving, work?

We had something similar when I was an enlisted soldier at Fort Hood, and it worked great. We had a Junior Enlisted Council, where every platoon within the battalion had a representative (and one of those representatives was designated as the Chair of the Council). The Council met monthly, sorted through all of the various problems, focused on those that were legitimate, prioritized those needing to be addressed most urgently, and then went to the Battalion Commander with proposed solutions. He fixed so many problems in such a short manner that, before long, the subordinate leaders (Company Commanders, 1SGs, etc.) were asking us to bring issues directly to them, and they'd fix it at their level (rather than having the big boss come down onto them). We solved issues with leave policies, duty rosters, unfair/uneven treatment of soldiers, harassment, etc. It can work, but if you have a responsive chain of command (who actually listens and takes action), it shouldn't be necessary.

This could be a good idea (if OPTEMPO allows for it), but you and I both know that Soldiers will just turn it into a b*tch session about anything and everything.

I think there is little possibility this would work. The military isn't good at doing things by committee, just not set up that way by culture or training. There are avenues for junior enlisted personnel to bring legitimate complaints to the attention of the chain of command. BTW, you might be surprised at what SNCOs and Field Grade officers know. A lot of things get discussed in staff meetings with the XO or Commander. Not everything can be top priority. Sometimes they choose not to address some challenges because of limited resources and time. The other thing military organizations sometimes do poorly is providing feedback when a problem is identified.