You need a new tyre because of the wear, and probably new brake pads if its been sitting around for a couple years.
I can't tell if the main brake wire goes through the bridge and to one side, or if there's a separate yoke. If its a yoke wire between the two brake arms, I'd also suggest a wire catcher poking out of that hole in the crosspiece (where the old fashioned caliper brake would have mounted.)
Here's the wire catcher on my bike. Its really just a bolt with a long spoon-shaped catcher. I've made the same kind of thing from a long thin bolt and three nuts. The idea is if the main wire lets go, the straddle cable/yoke cable gets pulled down by the two springs in the canti arms. This pulls the wire into the tyre where it catches and you do a sudden unexpected deceleration. Admittedly this is more important on the front than the rear.
Yes the observant will see that I have a more modern wire where the yoke is only on one side, and the other side is the main brake cable. However I still don't trust it to release right should things go wrong, and I still need a bolt for my mudguard.