I think it remains profitable because churners are a small percentage of their customer base. Even some of those who do churn may be tempted by a particularly valuable card to hold on to it and pay the annual fee (I have five Chase cards and have no plans to ditch any of them).

On the expense side, banks have to pay for:
(1) A sign-up bonus. Let's call this $500 given the earlier Ultimate Rewards card bonus with 50K points at minimum valuation of 1 cent each.
(2) A referral bonus. I would argue that most customers do not actually sign up through referral links, and Chase actually does well by usually offering the same bonus on its own page. This way it doesn't force everyone to use affiliates for the best deals. Let's average across the entire customer base and say it costs them $50 per person.
(3) More points with spend. I assume $24K spend per year as described below. So that's $240.

On the income side, banks can make a lot of revenue:
(1) Annual fee. Not all people will stick around that long, but some like the Sapphire Preferred are pretty valuable. Let's call it $100 a year after the first year (which is usually waived).
(2) Fees based on number of transactions. Banks earn a minimum amount per transaction. I don't know the specifics, and I know they split it with the processors. Let's call it 10 cents per transaction. I make about 1,000 a year on my Sapphire. So that's $100.
(3) Fees based on amount of transaction. Again, banks earn a percentage and they have to split it. I don't know the specifics. Let's call it 1% that goes to the bank. So if you spend $2K a month or $24K, that's $240.

With my numbers, the cost of the points is free; the merchant is paying them. But with breakage and slightly higher transaction fees, the bank is probably making money off of this.

As for the rest, after three years the bank has made $200 in annual fees and another $300 for charging 10 cents on each transaction. That's $500 total. In contrast, the bank only paid one-time fees of $300 to sign up the person.

Again, all of this changes if the card actually gets churned. But there are plenty of cards like the US Airways MasterCard (with 5K redemption discount) or Hyatt Visa (with free tier-4 hotel each year) that have benefits worth the cost of renewing. I plan to keep these cards. A lot of other people do, too. If a few people churn, it lowers the average profit margin, but not by a huge amount.