The point of bathing is to rub off dead skin cells, excess oil off your skin (note that I said excess). And you can actually do all that without soap.

Also note that sweat it's not that stinky when it evaporates quickly (for example, behind loose non-western clothes in a hot environment)

Soap helps a lot, of course, but the usual stench of being dirty is more because we use really tight clothes that keep sweat from evaporating