I'll give ya a little Handi and Winchester supreme experiance.
I have a 7-08 Handi, set it up for my son and WT hunting, the best factory ammo groups I could get was using Winchester Supreme 140 Silver tips, and yes they are black. They use the Nosler Ballistic Silver Tip bullet. What ever Nosler uses to coat their bullet with is a patent name Lubilox. Is that Moly? Who knows? It's some special formula they have and for all practical purpose it's a moly lube derivitive or simular enough.
When I went back to useing regular cup-n-core copper jacket bullets, like you've been told, accuracy went south! And yes, it was a booger to clean that bore till a white patch came out white. Now maybe there is some magic mix that will clean that Lubilox deposit easily, but I didn't have any. It was a couple days worth of swabbing, brushing, patching and waiting overnight for the cleaner I used to get the job done.
Once it was clean, shooting plain jane copper bullets started working again. It didn't screw-up my barrel, it's fine, but for me I'll stay away from the coated or lubricated bullets from this point forward. If you have 5 boxes go ahead and shoot'em if ya can't trade'm, it won't cause any damage. But there will be a super cleaning job waiting to be done when their gone.