Does oxygenating tap water remove the tap water flavor?
Many public water systems spray water into the air or trickle over rocks (aeration) to let some dissolved gases bubble out (e.g. odorous $\ce{H2S}$). This also may make the water less acid and thereby precipitate dissolved manganese and iron.

When tap water treated with chlorine, bromine, ozone or other disinfectant is allowed to stand, open, then those chemicals also can escape.

However, I doubt anyone actually tastes dissolved oxygen (the human body can sense $\ce{CO2}$, but not oxygen, which is why miners kept canaries. In addition, one can taste the acid of $\ce{CO2}$ in soda water, and feel the bubbles bursting.)

So the practice of letting water stand and refrigerating is useful in improving flavor, but $\ce{O2}$ has little to do with it. Chilling water does increase the solubility of gases, though.