Yeah, they'd lose, but probably more importantly, I doubt they care too much about Credit One.
If anything, Credit One being in the marketplace (with more exposure than they deserve) is probably a good thing for all reputable banks. Credit One offering garbage products makes all the offerings of other banks look even more attractive.
The Credit One logo might tempt unsuspecting people to open the envelope. Now, once they actually read what's in that envelope, they'll realize just how awesome Capital One actually is in comparison.
I'm still relatively new to credit cards, and it's only when the Credit One/Premier/Verve mailers come out that I'm reminded just how awesome it is that there are so many great rewards cards with no fees out there.
Credit One used to send me so many promotional offers in the mail. Years ago, I took the bait and accepted and then two seconds later a friend called with more credit experience and told me I had better cancel that real quick because Credit One is not an honset company, that the fees are very high, and that if you ever need help it is almost impossible to get customer service and that they are not very nice at all. So I followed her advice. I feel like a lot of young people fall for them. It's a genuine concern--better to wait out the time till your credit is more established and you can get a decent credit card company than to go with Credit One and have to worry about potential fraud (maybe even from their own company, since there are stories of that happening.), and not being able to get any help with your account if you absolutely need it.