I think there may be a legitimate problem with the gun, but there may also be something wrong with your choice of ammo.
I have a six shot 617 that will, with a scope, shoot 5/8" groups at 25 yards using Federal match ammo (the expensive stuff). It shoots 1.5" groups with Federal Walmart promotional ammo. It shoots most other high velocity hollow points with 1.5" groups.
The cylinder does have a little play, as do all double action revolvers, but it by no means sloppy.
Your ammo, however, is stuff that I have specifically had bad results with. The worst ammo I've used, which is not on your list, is the $11 per box Eley Red target ammo. The groups were about 4" at 25 yards. The super high velocity ones like the stinger also gave bad groups, as did Remington subsonic.
The ones that work well, and which I would recommend that you try are:
Walmart Federal Bulk pack hollow points (550 box)
Federal high velocity hollow points
Winchester Super X high velocity hollow points
CCI mini mags
Federal 711B standard velocity lead
CCI standard velocity
All of these are modestly priced, especially the Walmart's, which cost under $10.