That would be a chain, but if that broke you'd hear it and it would make a very loud clatter.
Unless they're talking about the sprocket that drives the water pump...have them give you the details as to what is broken and what will be replaced. Will give us a better idea of what they're saying is wrong.
Not necessarily. The balancer chain drives the pump. I'm betting one of the guides broke on the balancer chain but you would have major clattering noise from that. Chains don't usually break in the motor...they usually stretch and start to make noise, unless you have 200k on it and have neglected it. If this is the case, here's what I remember from work:
Timing/Balancer chain replacement: 18hrs. about $3000 - this would include replacing both chains and all sprockets, with the exception of the camshaft sprockets.
WP = 9.8hrs so about $1900 dependent on $$$/labor hr.
I'm in the middle of doing my chains currently and it's not an easy job. It's been 4 hours of my time just to get the top off, now the front has to come off....that's happening tomorrow.
On a lighter note, you can access the the front of the pump to see if the chain is still on it and if it has play and if the WP sprocket spins freely. Just remove the access cover in front of the pump. My guess is the dealer doesn't want to do it because the tech has to do it for free...unless a warranty company paid for it last time. Even then, some of them are sketchy and won't pay but for the part. Or the original tech is gone now....Been in the dealer world for 15+ years...seen it all.
EDIT: Just read you have to pay for labor, so scratch that tech not wanting to do it....