I've heard this argument before, but I'm not sure that the numbers support it. Despite the Dreamcast having a head start, the PS2 started eclipsing the DC's sales almost immediately, and that's even with the PS2 having some supply problems early on.
If piracy was the main problem, I would expect to see huge system sales and small game sales. Instead, the DC just didn't sell very well outside of its initial launch.
I'm not saying piracy didn't exist, but Sega had lost so much support from customers and developers with the 32X, Sega CD, and Saturn, I suspect those are more to blame. They'd have been able to handle the problem of game copying better if they didn't have a dozen other problems at the same time. Heck, it was the first console with built-in online services, and that's the industry's main way of dealing with piracy now.