External "lungs" that process your blood the same as your real lungs do - would that stop your need for breathing?

A device that fits your description does seem to exist, namely an ECMO machine that delivers extracorporeal membrane oxygenation (Fig.1 ). It is defined as:

[A] machine [that] pumps blood from the patient’s body to an artificial lung (oxygenator) that adds oxygen to it and removes carbon dioxide. Thus, it replaces the function of the person’s own lungs. The ECMO machine then sends the blood back to the patient via a pump with the same force as the heart, replacing its function.

The response of patients to ECMO appears to be variable (Crotti et al., 2017):

As the physiologic cardiorespiratory variability is increased in this condition, the dynamic interaction between patient respiratory activity and extracorporeal system function affects the clinical management. The effect of extracorporeal CO2 removal on patient respiratory drive is variable and not always predictable, with some patients responding to CO2 removal with a decrease in respiratory rate and effort and other patients demonstrating a persistently high work of breathing independent on CO2 unload.

Breathing and undoubtedly the exaggerated gasping for air after heave exercise is related to the concentration of CO₂ in the blood that's monitored by the brain. Those chemoreceptors basically measure the pH of the blood. However, pulmonary pH receptors exist and a multitude of other physiological mechanisms are at play to keep the level of O₂ in check . I happily refer to Brinkman et al. (2020) for an overview on O₂ and CO₂ homeostasis.

^{Fig.1. ECMO. source: University of Iowa}

<sub>References
- Brinkman et al., StatPearl
- Crotti et al., J Thorac Dis. 2018 Mar; 10(Suppl 5): S661–S669</sub>