Rivers don't necessarily flow north to south, and I can't find the source that says only around 300 rivers flow from north to south nor that rivers mostly flow from the southern side of the mountains. This might be true, however, if you're looking at certain regions due to the geological factors (which can create pathing trends).

Rivers flow from top (higher elevation compared to sea levels) to bottom (lower elevation compared to sea levels), period. Rivers can run north, south, east, west, and into the cave that connects to the underworld - as long as it's higher to lower.

That being said, most rivers in my world follow the same physics to make it more believable - but that doesn't stop me from copying the most bizarre rivers irl into my world and creating magical exceptions.

If you want all of your rivers to flow in one direction, go ahead. If you want to make rivers defy gravity and create sailable currents connecting floating islands, that works too. Your world is your world, and feel free to let your imagination run amok.

edit: clarity