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....

The box should not wiggle, loose things just get looser. Confirm it is actually a ceiling fan box. Confirm it's actually mounted to the joist. Make sure the correct fasteners were used to fasten it to the joist. Replace or tighten the fasteners. If there's not a joist there then the bracket you posted is a good option.

In my experience with Hunter fans the ceiling fan hanging bracket does not depend on the plastic box for support. Two long, special, wood screws come as part of the fan hardware; they are used to secure the fan hanging bracket through HOLES in the plastic box into a ceiling joist or a 2x4 or a 2x6 cross member going between two joists.

I have never seen a bracket with this metal strip. Perhaps, that steel strip is strongly attached to a secure member and you can attach the fan hanging bracket to the steel with suitable bolts. Or that strip may have been part of the hardware of the original fan and not be needed for the new one.

You don't have to remove the box to determine whether you have a suitable wooden support. With a screwdriver or a wire feel around between the outer edge of the box and the sheetrock to see what you have. If the new screws don't tighten hard, then you could insert anchors (but this might require drilling out the holes to fit the anchors).