Why are there no Aarti near river banks in South India? Is it that South Indians do not worship rivers?

 There is no need.

South Indians, especially Tamils, standardised rituals (by 63 Nayanmars, 12 Azhwars, Sankarar, Ramanujar during 8–12 century and Arunagirinathar during 15th century) in Temples which will take care of many things including Water element (Theerthams).

You may find one or many theerthams inside the temple complex and rituals are done 3–6 times, a day.

Look at the banks of Cauvery and Tamirabharani- innumerable temples on both sides of the banks, a big temple in a small village is common which shows its affluency till 17th century, before Marathas, Nawabs, British and French invaded.