An air eliminator is always a good idea but don't confuse O2 migration with air. The O2 will diffuse in the water and bond to whatever sediment or particulate that floats around the system. Air will raise up to the highest point to be vented (hopefully)
I don't get the desire to not buy O2 barrier pipe. Tubing is cheap these days and if it is the difference of $.05/ft, what should it matter. It is extra protection.
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