Economically speaking, non-refundability doesn't necessarily make overbooking unnecessary.
Let's take two airlines, A and B. Both only sell non-refundable tickes. A never overbooks. B always overbooks. What happens?
A will fly with some number of empty seats on average, because they still get the occasional no-show. B flies with fewer empty seats on average because they overbook.
Which airline can offer more competitive prices? It's B. Because by overbooking B's revenue per plane leg is higher.
If that isn't clear, here's an example. Say the planes have a capacity of 100, and on average 2 people fail to show. A sells 100 tickets for $100.00 each. B sells 101 tickets for $99.02 each. On average, B makes more money on every flight, _and_ is more competitive on price.
Of course with the numbers in my example the compensation paid by B on the odd occasion that everyone shows up probably kills any additional profit. The real numbers matter in order to determine to what extent this really happens in practice. But I hope I've demonstrated that non-refundability of tickets doesn't necessarily or automatically mean that overbooking is not profitable.
So there is some potential level of competitive benefit to overbooking even on non-refundable tickets, assuming that some people still occasionally fail to show (because they miss a connection, become ill, or whatever).