I'm pretty sure 90% of the Wii userbase has no idea that the Wii even plays Gamecube games and might not even know what the Gamecube is. Anyone who cares already has a Wii, so it's not really a big deal. Nintendo never drew much attention to the backwards compatibility in the first place. I think they've learned that if you don't have it you can resell the old games.
Backwards compatibility serves a purpose in the first few years of a console's life by helping to win over the previous console's user base. Not that the GC had a large user base, mind you, but it was a good thing to have from the time the system launched until about let's say 2008. After that pretty much everyone who cared about that feature already owned it, so Nintendo reasonably could have removed it at that time. I'm surprised it took them this long to cut it out.
I doubt it added much to manufacturing costs, but even if it was only like $5 you figure Nintendo may sell another 20-30 million units, so that adds up to a significant chunk of money saved off manufacturing. It is also highly beneficial to reduce the size and weight of the Wii as much as possible, and taking GC compatibility out was a logical way to achieve that. Not that the Wii was a large or heavy system to begin with, but getting it even smaller is nice.