They shouldn't overbook a plane, and things like this don't happen.
It's not that simple.
All scheduled airlines overbook. They know that there will always be a certain percentage of no-shows so overbooking allows them to operate with a higher average 'load factor', i.e. a higher percentage of the seats occupied. That in turn allows them to offer lower fares, which is what most customers want more than anything else and which is evidenced by the success of 'no-frills' carriers.
And over booking works. In the 1990s, major airlines typically operated with load factors around 65-70%. These days, it's typically around 80-85%.
So if one airline decides not to overbook, immediately it's giving its competitors a price advantage of around 20%. That's commercial suicide.
Absolutely! "If" people just don't turn up, I assume they keep the fare.
So the seats are paid for, that's being greedy trying to get double bubble on the seat
It doesn't work like that.
Some people are no-shows because they have flexible tickets which don't tie them to a specific flight. So they'll have a seat reserved on one flight, but if the business meeting over-runs or something else happens they'll switch to a different flight. They've paid handsomely for that privilege, but at the end of the day they've paid for one seat and they get one seat, so no double-counting.
Of course, some people who are no-shows have tickets which are restricted to that flight and won't be getting a refund. But that effectively cross-subsidises and reduces the price of the tickets for the passengers who do show up.
You can't accuse airlines of making excess profits because they don't make excess profits. In fact, most of the time most airlines don't make any profit at all.