It's not true.
I thought it was also, so I looked at United Airlines 2024 Annual Report to confirm or deny my position. [^0]
If you look at their revenue sheet, you'll see that UAL made ~$51B from passenger revenue (tickets sold * available seats across their entire fleet) while they made ~$24B from "other revenue", which includes, amongst other things, annual fees from credit cards.
Same with Delta, though they made ~$10B from their "other" revenue. [^1]
However, it's a bit of a positive feedback loop situation. The "other" revenues in these 10-Ks don't tell the whole story.
Airline frequent flyer programs have tiers with minimum flight and spend requirements per tier. This benefits both frequent flyers and VFR customers (visiting friends and relatives).
If you travel a lot for work (frequent flyer), there are very heavy incentives to get to that top tier. Customer service at the highest tiers is eons better than what you'd get at lower tiers. You also get priority boarding, first-crack at upgrades, upgrade certificates that move you to the top of upgrade lists, and more. These benefits make air travel, which many people don't like doing, much more tolerable.
If you're an infrequent flyer, getting to that airline's mid-grade tier usually gets you more free checked bags and priority boarding. Checked bags are EXPENSIVE after the first freebie (thanks, Southwest!) which is usually enough of a draw to get people to chase that status. (If you live near a hub, you can gamble and hope that the gate agent offers to check bags gate-side for free to speed up boarding, but that's not foolproof. Anyway, checking baggage is a fool's errand; one-bag for life!).
Getting the airline's co-branded card usually provides bonuses that make it easier to hit those tiers. So you get the card and put all of your personal (and corporate, if your company allows it) expenses on the card.
Airlines also have gotten very aggressive about pushing the card onto gen pop. You're almost certainly going to get hit with a 60-80k mile offer on every flight you take in the US for spending ~$3k on that airline's co-branded card, no matter the airline. (It's almost always enough for a round-trip ticket to some coveted location in the US, on an award flight, which are harder and harder to come by, but that's another topic for another post.)
United flew 173M customers in 2024. $3k card spend from even 10% of those customers is $52M! And that's before you consider that most people will continue spending on credit cards after earning the spend benefit! (However, at $0.01/mile earn rate, the $14M worth of flights United would be beholden to is recorded as a "frequent flyer deferred revenue" liability. But, again, the chase for status and benefits would generate more revenue that's hard to forecast, though I'm sure the airlines have forecasting models in place.)
If this interests you, and if you like math, "The Global Airline Industry" by Belobaba et. al. is a fantastic book that explains this and other peculiarities of how airlines work. This was recommended to me by an old colleague that ran a small airline. It's excellent.
[^0] https://ir.united.com/static-files/d4c854c7-427c-49a9-8129-d...
[^1]: https://s2.q4cdn.com/181345880/files/doc_financials/2025/q4/...