(Not an expert on ventilators, but I have been working on software side of medical devises.)
Producing semi-mechanical Bag-valve-masks (BVM) would be relatively easy, but you need professional to sit there and monitor the patient. That's not ICU ventilator. BVM is very crude tool for short term use. If the rules and legislation are relaxed so that hospitals could use non-certified crude devises that help. You would almost certainly see ventilator-induced lung injuries.
Ventilator-associated lung injury https://en.wikipedia.org/wiki/Ventilator-associated_lung_inj...
Ventilator-Induced Lung Injury Review (Part 1 of 2)
https://pulmccm.org/review-articles/ventilator-induced-lung-...
If the system is automated life support, it generally has to maintain homeostasis of the patient and _dynamically_ adjust. If you are just tiny amount off from the optimal and there is drift in the dynamical adjustment it accumulates over time and you can kill the patient or cause immense pain or discomfort.
Oxygen is poison if you get it too much and if you get it too little, you die. Pushing air into the lungs can damage them and the oxygen mix should adjust. If lungs are filling with fluids, you might need more oxygen and less volume. The system must be connected to instrumentation that monitors the patient's state continuously.
There is the question of medical staff using the thing correctly. Medical UI/UX is safety critical in the same level as it is in aviation. If just 5% of users use it wrong it, can negate all benefits from using the device.
Generally you need deep knowledge transfer to scale up production of critical components or testing and calibration, just emailing the specs is not quick enough. You need to test it with real environment and refine the device to see how it works in practice. I'm sure that ventilator manufacturers have already outsourced every component that is easy to manufactured and produce to others. There are some bottlenecks that involve testing, special machines, materials and calibration that is not easy to scale quickly.
(edit: the time-frame of ICU overflow seems to be between May-Sep in the most countries in most models If you ramp up deliveries 3 months from now, you could catch the tail end of the need)