I have several sets of RCBS, Hornady, one set of Lee. Never had a Redding set hear alot of good things about them and understand they are high quality...however..Several months ago started looking at upgrading press, dies, etc..it came down to either Redding or Forester for me. I ended up choosing Forester..largely becuase of the Co-Ax Press. Now when possible I purchase Forester's product, although Redding would be next choice..If I had purchased a Redding press I probably would have done that way on dies. The only downside to the Co-Ax is some of the larger dies will not work...so I have to use the RCBS press I have on those.

The die locking rings are much better in my opinion...I have changed out all my dies to the Forester rings ...I like the easy of use and it is solid in my press...much better than the other systems for dies lock out...You may want to try a set to see if you like them..

I have RCBS, Forster and Redding dies that I use all the time and I've put Hornady lock rings on all of them. I use the Forster ultra benchreast seating dies and have to look for runout with a dial indicator.

I use mostly Redding dies and a few of everyone elses. One thing I always do is keep a supply of Hornady lock rings and replace every dies rings with Hornadys.
Read this as - I think the Redding die is best and that Hornady makes the best lock ring. I'm not a huge fan of Hornady dies though.

I always square my dies like you, but have never had a problem with the runout changing or the lock rings loosening. I use mostly Lee and Forster dies, but do have some Redding dies. I have no problems with the design of the Redding ring. I hand tighten the Redding and Forster dies when squaring them. With the Lee's I use an RCBS die wrench to tighten them down all the way. Many people do not like the Lee ring because if you hand tighten them, they are not locked 100% tight because of the rubber O-ring (and some people are against using a wrench). You can take the O-ring out if you want to.
To answer your question, I think any ring should lock your dies adequately if they are tightened enough. My only other advice would be to also check your press, including the shellholder and the clip in the ram that locks your shellholder in place. This clip can keep your shellholder from sitting flat on the ram which can sometimes make it "wobble" when reloading. Some people take it out and use a rubber o-ring to keep their shellholder in place (and to let it self center). Keep trying and don't stop squaring your dies.