While the ultimate purpose of breathing could be considered to be the maintainance of a balance of the substances you are referring to (such as blood oxygen, carbon dioxide, and hydrogen ions), the blood levels of these substances do not directly control the production of action potentials within the motor neurons that promote the contraction of the diaphragm and intercostal muscles.
The propagation of these action potentials is initiated by signals from the medullary respiratory center, specifically the neurons in the dorsal respiratory group (DRG) and the ventral respiratory group (VRG). In the VRG, a complex of neurons known pre-Bötzinger complex is responsible for generating the signals that cause the rhythmic muscle contractions involved in breathing:
The respiratory rhythm generator is located in the pre-Bötzinger complex of neurons in the upper part of the VRG. This rhythm generator appears to be composed of pacemaker cells and a complex neural network that, acting together, set the basal respiratory rate.
Vanders Physiology, p473, 15th ed.
So breathing is indeed, as you mention, a process that is controlled by innate neuronal activity but regulated by the concentrations of PO2, PCO2, and H+ concentrations. I recommend you read pages 473 to 477 of Vanders Physiology, which explains these controls in detail, some of which do involve reflexes, say, if O2 concentration in the blood strays too low.
Interestingly, expiration, which is typically a result of actions potentials ceasing and respiratory muscles relaxing, can be controlled by a reflex, known as the Hering-Breuer reflex, during strenuous exercise when the lung is inflated by a large tidal volume. Stretch receptors in the airway in this case are activated, causing the inhibition of inspiratory neurons in the DRG.
Source: Vanders Physiology, 15th ed, section 13.9: "Control of Respiration"