Second of all, you're talking about gargantuan quantities of oxygen (or nitrogen, or whatever). CO2 is roughly 1/2500 of the atmosphere, so to achieve a given amount of dilution, you can remove one CO2 molecule or add 2500 other molecules.
Finally, I don't know what the side effects of increasing global air pressure would be, but I'm sure there would be significant effects!
Anyway, as you said, any project that doesn't involve significantly scaling down anthropic GHG emissions and human settlement footprint will have significant effects. Planting trillions of trees will result in dead trees and environments destroyed. CCS at scale will involve ungodly amounts of energy + materials. Algae farming will alter the ocean's biodiversity already affected by warming + acidity + overfishing.
It's time to acknowledge that the Earth doesn't owe us a couple of large cars per household, exotic foods, meat BBQs and regular plane trips.
Of course all of this is academic in any event; addressing climate change by diluting the CO2 is a nice attempt at out-of-the-box thinking but it's not actually plausible.