Bearing are full seated into frame.
Not to sure if the re-tensioning nut in loose or not but I’m guessing it must be but not sure how that can be.Is it worth a bit of loctite on it to see if it helps?
Once you have tightened the black tensioning nut, there should be no way that play can develop unless that nut comes loose (or grossly deforms). You are creating a fixed distance between the crank arms when tightening that nut and if it’s not coming undone, then I am not sure how that distance can increase to generate play in the crank.
I would take the crank arm off and make double sure that the crank is fully inserted. Give it a tap on the drive side with a rubber mallet if need be. When re-fitting the non drive-side crank arm, the tensioning bolt will drag the axle through if need be.
I would then mark the crankarm/black bolt so you can see if it is coming undone – might give you a clue about what’s wrong.
Probably a stupid question, but you are definitely pre-loading the bearings with the black bolt, then leaving that in place and tightening the crank arms? And you are leaving the black bolt in place once the two crank arm bolts are tight?