This is incorrect, or at least incomplete.

You are correct in that it isn't recommended to give rescue breaths for untrained people, or even trained people who are rusty. Trained first responders DO give rescue breaths. There are a few reasons the general public is advised not to give breaths:

• There's a certain subset of the population who just won't give CPR at all if they think rescue breathing is required, and compression-only CPR that they are willing to do is infinitely better than perfect CPR they will not.
• Most people suck at both rescue breathing and chest compressions, and time spent poorly cycling the air in their lungs that is still mostly oxygenated because their blood hasn't been able to circulate well is time better spent trying to circulate it better.
• There's a risk of transmitting disease without certain equipment that bystanders aren't likely to have on them.

That said, trained first responders are still taught to give rescue breaths, because when done properly, it safely enhances both survival and recovery rates.