I agree, "I swim on Saturday" is an acceptable (although, possibly not to prescriptive grammarians) form of the idea. "Hey,you want to play ball one Saturday?"
"No, I can't. I swim on Saturday."
Definitely gets used, even if it isn;t "correct"
Not sure if I'm right, but from what I think I understand on Saturday would be "de Sathairn" (if that's the correct spelling) and on Saturdays would be "ar an Sathairn". That seems to be the pattern that I'm seeing.