Bergs, very common across the crinkle hub series. I know from experience, thought the end of the world had come.
A new splined drive ring in the drum will remove some of the backlash, but the way you ride there is no cause for concern here because the driven faces will be under load for 99% of the time, and a small clunk on the over run will be lost on the wind. Mine clunks a bit, but the rubber ring keeps it reasonably quiet. However, excessive wear on the splines will allow the drum to rock laterally, as close fitting splines offer additional support to the back face of the drum, less support with loose splines. Check in the same way you check for wear in track rod ends on your old motor, by grabbing the drum, front and back, see how much it rocks.
More interesting is the wheel hub. It is a basic cylinder with the two crinkle pressings riveted or bolted to it. Flexing of the parts cracks and shears these fixings. If the sideways rock is not visible outside the hub, the bearings or bearing locations are at fault. A small movement at the hub feels bigger at the rim.
The options are a different wheel, or to repair yours by stripping down to the bare hub, vee out the centre crinkle join and weld the two crinkles and centre tube together. In manufacture the crinkles were pressed onto the tube, and over time this tight fit loosens allowing slight movement between the parts, fretting the holes, loosening and shearing the rivets.
The rivet or bolt holes can also be used to plug weld on to the centre tube. With a bit of force and luck it may be possible to hold the centre tube still and rotate the crinkles equally until the through holes are covered, allowing good plug welds onto the tube. Then put it in a lathe and turn a nice flat bottomed vee just down to the tube between the crinkles and weld the three parts into a good solid unit. This is easier than trying to replicate factory rivets, and a more effective repair. A whole lot stronger than original, and with the welds smoothed off, job's a good 'un.
Swarfy.