Oxygen is a relatively light gas, and at 1 atm and room temperature the root-mean-square speed of each molecule is almost 2 kilometers per hour.
If you are oxygenating a planet that already has a ~1 atm atmosphere, you will get a higher local concentration of oxygen around the device.
If you are making an earthly atmosphere out of a near vacuum such as the martian atmosphere, you will only get breathable air after the whole planet has stabilized at a breathable pressure. If you do that from just one point at the pole you will either take millenia, or, if you do it fast (and bh fast I mean doing it in decades or centuries) you will have winds that will make colonization and forrests impossible. You'd have a better time seeding an atmosphere from multiple points in the planet.
If you start with a venusian atmosphere, first you have to lower the pressure by ~89 atm. Good luck with that.
Other than that, the Earth itself has had a lushing forrest on the south pole, back when dinosaurs roamed the planet.
Go back 100 million years ago and Antarctica was covered in lush rainforests similar to those that exist in New Zealand today," said Dr Vanessa Bowman who works with Francis at the University of Leeds.
"We commonly find whole fossilised logs that must have come from really big trees."
I totally stole the link above from a comment for another answer.