If a PCB fab can't reliably deliver boards with intact copper artwork when your layout complies with the minimum widths and other design rules that they specify, they don't deserve your business IMO. If they can't reliably do 6/6 they shouldn't advertise it.
I've never had a problem with 6 thou from OSHPark, but I've never used the other one you mention. But 6/6 is nothing to worry about for any decent PCB house, and there are lots of low-cost options around the world that are decent.