Physiotherapists and oxygen saturation

Just a question to everybody about a practice that I see commonly on the ward but can't seem to find any evidence of benefit and it really irks me.

Old person, resolving chest infection or the like, now off oxygen, trying to get him/her home. Physio stops walking him because he desaturates on exertion and attaches oxygen (increasing difficulty of walking independently tenfold and prolonging hospitalisation). Alternatively patient already on oxygen, stopped from walking because of desaturation on exertion making them "unsafe to mobilise".

I frequently request that physio don't measure oxygen sats on exertion for these patients as frankly I don't care what the number is. There are plenty of chronically hypoxic people in the community functioning perfectly adequately with sats in the low 80s when they go for a brief waddle off to the toilet.

Is there any evidence that obsessing so much about the minute-by-minute oxygen saturation on exertion correlates to anything meaningful in somebody recovering from an acute insult? And from that, is there any evidence that tethering these people to portable oxygen canisters actually does anything?

Thanks