Replacing ceiling fan bracket
Assuming that this box isn't tied directly to a joist, then yes, a replacement bracket like you've shown could help.  Either way, you should take off the shiny bracket and confirm how this box is mounted, and check if it is sturdy or not.

However, most "wobbles" are caused by blade imbalances, not from the mounting box.  The faster they run, the more susceptible they are to wobbling.  They do have balancing kits for sale at local places usually.  They use peel and stick weights, and a slide-on clip, to help figure out where the peel-and-stick weights go on (on the side towards the ceiling of course.)

The paint "melting" look is from a previous fan being in place, and years of repainting around it got it stuck to the old escutcheon/cover plate....