My understanding is that the O2 sensor's main goal is to get the fuel mixture as close as possible to the stoichiometric ratio of 14.7:1 so that the catalytic converter can function at its peak. But without considering the needs of a catalytic converter, can an engine run properly and be efficient by using other sensors readings like the Maf/Map sensor
Before fuel injection and catalytic converters, carburetor engines ran without many (or in some cases, any) electronic sensors. They'd use mechanical springs, grub screw adjustments and vacuum pipes to keep the fuelling in check.
I should probably add that the reason for the O2 sensor was to protect the cat from getting drowned in unburned fuel which can happen occasionally in a traditional carburetor car. That said, carburetor based cars were capable of returning good economy and performance, in most cases within 90% of fuel injection models.
While the answer by Steve Matthews is good, I have something to add to it. Consider this: if the O2 sensor fails, you can still continue driving the car. Yes, you will get a yellow check engine light but it will not prevent you from driving the car.
Emissions will be worse with a failed O2 sensor (because closed loop control of the process is impossible so there will be either too much HC and CO or too much NOx depending on whether you have a lean or a rich mixture).
So, based on this, the car can continue operating on the other sensors alone such as the air flow sensor. However, if both the air flow sensor and the O2 sensor fail at the same time, you may be in trouble (unless the car has a real throttle position sensor and not just an on/off idle switch, but that can fail too... -- modern cars typically have a real throttle position sensor or two as they have an electronic throttle control system).
There is plenty of redundancy in the sensors in a modern car. Unfortunately, failure of crankshaft position sensor may cause the engine not to run, so a single failure can be problematic.
The reasons for introduction of oxygen sensor in cars are less pollution to the environment and better fuel economy.
When a car is brand new everything is adjusted so the air/fuel mixture is optimal without using the O2 sensor. In time, due to wear of parts, residues of liquid and/or tar deposit on surfaces, sensors becoming less accurate, etc. the air/fuel mixture becomes incorrect to some degree.
The closed loop circuit with O2 sensor makes corrections of air/fuel ratio based on readings from O2 sensor. This circuit is not essential for proper engine functioning. The closed loop continuously adjusts the air/fuel ratio and and makes it optimal in all engine modes. If the O2 sensor is faulty or disconnected, the closed loop circuit is switched off and everything defaults to settings like in a car which does not have a O2 sensor at all.