Yeah. Effective compressions are more critical right away, especially if you’re alone. There’s some passive ventilation happening with the compressions.
Breathing IS part of effective cpr. It is absolutely recommended. Compressions alone are better than nothing, which is why we had a brief period where that was taught for one-person-cpr (and was meant to make nonmedical personnel more likely to do SOMETHING), but anyone trained in cpr should absolutely be following the compression:breathing ratio taught in cpr classes.