remedying ceiling fan horror-show
TL;DR: That is a Hunter Original Hanger Bracket, which consists of a rubber bushing and U bolt rocker assembly. Your option #2, drill box and attach through, is the correct installation: you just need a drillable box.

That hanger design is (a) meant to work with an outlet box and (b) to be mounted directly to the joist. Huh? How does this work? Here is an image of the original installation instruction:



Per those instructions, the installer is meant to drill through the box, attaching the lag screws into the joist behind the box. Modern boxes may not accommodate that drilling, but I believe an older Bell round box would work:



If you want this code compliant for today, you can't alter the box, so you can't install this as it was meant to be: thus replacing the fan is the only compliant choice. If you want something safer, but not modern code compliant, hunt down a vintage box or have one fabricated or "just make one work": the important points safety-wise are (a) getting the wires in a box and (b) securing the fan to the joist.

Those are solid fans. I would keep it if possible.

Note I am not affiliated with either of those eBay sellers linked to the images.