Our muscles do run off phosphorus.
Adenosine triphosphate (ATP) is what cells use for energy in all sorts of processes, including muscle contraction. The high energy phosphate is a handy way to store energy and release it into chemical actions.
You see that ATP gives up a high energy phosphate bond to "cock" myosin in muscles, getting it ready to contract. On giving up the phosphate the ATP becomes ADP (diphosphate instead of triphosphate)
We need oxygen to regenerate ATP from ADP. When fuel molecules (reduced carbon, like sugar or fat or ethanol) is oxidized, the energy released is captured by regenerating ADP back to ATP. The Krebs cycle shows exactly how that happens if you are really digging the biochemistry.
As far as the prospect of using phosphate as an electron donor instead of oxygen, the problem is that there is a lot more oxygen around than phosphate. Plus oxygen is volatile which makes it easier to breathe than phosphate, which would would have to drink in solution. But what about places where there is lots of phosphate? Could you use it instead of oxygen? I found this:
There are, however, numerous reports of trace levels of a reduced form
of phosphorus, phosphine (PH3), being present in certain environments.
The Wil-O’ the Wisp observed at night-time over peat bogs, swamps,
cemeteries, recent battle fields and stagnant waters may not in fact
be due to manifestations of the supernatural, but to the spontaneous
ignition of gas containing traces of phosphine which evolved during
the decomposition ofanimals in damp soils (Mellor 1940). In the 1920s
a debate raged about whether thebiological reduction of phosphate
really occurred (Rudakov 1927, Rudakov 1929, Liebert1927) and this
argument has not yet been resolved.
My pubmed said it is still not resolved. Unclear what exactly the microbes are doing with phosphate and why they generate phosphine.
The cool thing I learned is that will-o-wisps are associated with recent battlefields! Pretty creepy, but makes sense - our bodies are loaded with phosphates.