Ceiling Fan

Unfortunately, I am very much a mechanically challenged person. I am great with electronics, but mechanics just aren't my game. I'm trying to determine if my ceiling boxes are ok to use for ceiling fans. I am rather particular about wobble, noise, shake, etc so I wanted to see how best to be sure before I go ahead and install.
First, my house was built in 1959 in WI and I am not the original owner. When I moved in there were ceiling fans in each of 3 bedrooms. All the fans have some wobble to them. One wobbles terribly. They all appear to be cheaper fans with a couple geared towards kids - ie, the crayon, crayola, and multicolor fans. These two crayon fans are ones I want to replace as I have no kids and they just aren't my style.
I wanted to investigate the mounting and boxes so I removed the blades and housing on one fan. It just lets me see the rim of the box. If I rock the fixture back and forth it moves considerably, but the ceiling boxes moves only slightly. It appears that this one is loose screws. I would say that roughyl 99% of the movement is the fan itself and 1% is the ceiling box. Before taking this one down further, I wanted to compare it to the other crayon fixture.
I removed the 2nd fan to the same point and it too, moved when rocked with 99% of the movement seeming to be the fan itself. I took this one down all the way. The ceiling is round and metal. There appears to be a round ring mounted to the face of the box and this is what the fan was secured to. This round ring is screwed to the face of the ceiling box from down up - meaning that its screw heads are under the drywall ceiling and not accessible from below. I suppose this means either the box was in before the drywall, the drywall was redone, or the round ring was on the box then mounted - the point is that one could not attach this ring once the box is mount because you can't get to the screw heads. I see one screw in the center of the box which appears to be securing a metal clip. I can see either a metal brace or metal conduit through the holes in the box above it. The ceiling box is made of metal and has three conduit runs coming into it with electrical. Pulling on the box from below really doesn't give much movement. Pushing up on the ring doesn't give much movement. Pushing up on the inside of the box gives a little movement - maybe a couple 32nds of an inch.
Does this seem like how solid one would expect a fan box to be? Any thoughts on whether this description sounds like a fan box already? I am a mechanical idiot so removing and reinstalling a new box would be a major project for me and if this sounds like what to expect from a fan box, then it seems that I should leave it as is - my only reason for concern is that in replacing these fans with something more neutral - wobble and noise is really annoying to me so I don't want the boxes and mounts to be the reason for that. It seems to me that these two are just loosely mount and lower quality fans and that's the cause more than this slight movement of the ceiling box. Can anyone add some details that might help me out?

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