Having worked for AA for a couple years as a gate agent, I can tell you this is not the practice at American. And I would think a passenger might have cause for serious complaint if other airlines actually use this factor to bump.
If you do not have a seat assignment on an oversold flight, but have checked in on time, you aren’t going to be arbitrarily bumped. The standard operating procedure during an oversold flight is to solicit volunteers to give up their tickets on that flight in exchange for a seat on a later flight and a travel voucher to be used towards the purchase of another ticket.
Flights get oversold because they have a history of a certain number of “no-shows,” and usually, no volunteers are needed to take a different flight. If everyone shows up for the flight, and noone will volunteer, then AA’s procedure (outlined in your tkt paperwork) is to bump the lowest fare ticket, provided it is not an unaccompanied minor, a passenger needing special assistance, or an older person. Unless it gets to this point, you won’t be bumped if you show up on time for your flight.
I ran into several folks who would book a flight because it was overbooked so they could volunteer. They would try to rack up as many vouchers as possible for future vacations. Not a bad deal, if your travel plans are flexible.
Sorry this got all rambly, but one of my pet peeves while working at the airport was having to deal with a passenger who was sure the all-powerful agent could erase them from the system at will. We didn’t have that much power.