You can just add the garlic to the oil a few minutes before you plan to consume it: Crush or mince the garlic, add it to the oil, whisk for 30 seconds, and pour through a strainer (ideally using a wooden spoon to squeeze the garlic against the strainer). This provides plenty of garlic flavor, but insufficient time for problematic microorganisms to multiply.
See my sibling reply for a clarification; I'm asking on a tangent about whether oxygenation would be helpful in room-temperature oil extractions generally; not whether it'd be helpful in room-temperature oil extractions for culinary use (which... aren't a thing.)
Because you might be trying to extract something that would spoil/lyse/denature under heat or low pH. Doesn't apply in this case, but I can think of lots of cases where it would.
I would bet that there are many Volatile Organic Compounds used for e.g. perfumes, that have to be industrially oil-extracted "delicately", to preserve their integrity while also keeping them dissolved. In these cases, traditional workarounds to doing a reaction at high heat (e.g. low pressure) would lose the product (because e.g. you need to off-gas a product of the reaction through a gas scrubber), but under low pressure the VOC would also vaporize and be lost.
If those compounds come from live tissue, e.g. from a flower, then you might get botulinum toxin in your product as the flower's tissue degrades anaerobically in the oil. But using a hyper-oxygenated (and non-organic) oil could prevent this.
Another use-case for room-temperature hyper-oxygenated extraction would be for extracting proteins in a recombinant chemical production process, where the produced proteins aren't expressed into the medium, but are retained inside the cells of the producing bacterial culture (and so you have to lyse the cells to get the proteins out, while not damaging the proteins in the process.) In many cases, the resulting protein would be intended for direct blood infusion into humans — e.g. recombinant Human Growth Hormone. So you wouldn't use oil (not suitable for blood), but nor could you use any solvent that would denature proteins. Likely, you'd use water with a very specific osmolarity — enough to pull water out of the cells until they autolyse, but not enough to harm the proteins. And again, you might get botulinum toxin as a side-protect here, where it'd be even harded to distinguish by fractionation from the intended protein. So you want to maybe ensure that the medium is aerated. (Or possibly use a bacteriostatic medium, if you have a follow-on process that can guarantee full separation of the bactericides from the product.)
Can you bubble oxygen through oil in a way that would keep aerobic bacterial reactions going, without making the oil go rancid?