Card companies are far more likely to look for new ways to get fees out of the folks who don't pay in full each month. And the competition for those that do pay in full will increase.
The new Schwab card with no fees, 2 % cash back on everything and first-class (eg. US) service, means it costs me 2 cents every time I charge a dollar with another card to earn a FF mile (1.6 cents w/ Starwood Amex, assuming a 1.25 airline conversion). At that cost, the value of getting a FF mile I will use in 4-5 years, with capacity limitations, etc., etc. is of very questionable added value.
The card companies would be wise to see that the cards they hold aren't nearly as good as one might think.