Here are two facts and one question [actually more than one question]:
Fact: we don't have enough ventilators in the world for everyone getting sick.
Fact: people will die because they don't have help breathing.

If necessity is the mother of invention who is the father?

Is it possible to make something like a manual hand pump or "rocking chair pump" that will push and pull more air in and out of the dying patient's lungs than what they can get themselves - but, importantly, not too much air to make their situation worse?

I am not a doctor or an engineer - but I am able to think outside the box and pose new questions that smarter people are able to answer.

It is normal that all the technology developed for respirators have set the rules - set the bar. These high-tech wonders are saving lives. This is the paradigm we naturally use: Respirators look like this. Can we explore a paradigm shift? What would a paradigm shift look like?

In most cases when we breathe we could take more air in and push more air out on any given breath. There is a safe medium, and our bodies naturally control our breathing rate depending on the need for Oxygen.

Could we temporarily, manually boost the air-flow either for ourselves or for another person without it becoming a danger?
If the answer is yes, a follow-up question could be what might this look like?

Should we assume an infected person would be too weak to pump their own manual respirator?
If someone is pumping air for another, helping them breathe (admittedly this might be an outrageous idea) could there be three pumping types: more air, less air, stable air - and the infect person communicates their need with one arrow - pointing up for more air, down for less air, and across for continued rate?

The device could have a max, min lung capacity to ensure nothing gets damaged.

The impetus for me writing this is a news report that some nursing homes might not send any of their infected people to the hospital - with the logic: "they'll all die anyway". Is there another way? Can this necessity mother some sort of invention?