Your sort of correct...but not quite.
Anaerobic life forms can survive without oxygen. Not neccessarly without air. In fact some will die from contacting oxygen.
But, yes some life appears to live without air such as fish. But in truth they extract their "air" from the water...as an example.
I think you will find that the bacteria that produces the botulin toxin extracts it's "air" requirement from microscopic gas pockets in the garlic. However, you are correct that the bacteria is anaerobic and as such covering with oil will block the oxygen and allow the bacteria to thrive to toxic levels.
However, you will find that (what I call normal hygiene) proper hygiene and refridgeration will stop botulism in garlic oil. Stay clean and keep your oil in the fridge and you should be fine.
PS...A few months ago I actually tried to make garlic oil and keep it at room temp. I fried the garlic in oil until the garlic had absulutely no water content left. I then strained it and took the oil back up to temp. Then bottled in very sterile bottles...You guessed it, it didn't work. Within a few days a white cloud of bacteria had grown.