Do the 'effects' of meditation happen because you relax your breathing and put yourself into a mild state of oxygen deprivation?

No, that's not the case. In normal meditation, your oxygen levels are fine -- the lower level of activity (and typically, anxiety) are consistent with reduced oxygen intake. I would agree that "chi" and "energy" are not sound ideas for understanding meditation and its effects. There are rational, non-magical-thinking ways to understand what's going on during meditation, both from a psychological and physical perspective. But, over the centuries, much of the explanation has had non-scientific characteristics because the people doing the explaining didn't have those tools available.