I see no reason why you shouldn't be able to continue paying interest-only until you die, at which point your house gets sold to pay back the lender
But that's not how banks work. They have set amounts of funding to lend on a certain batch of mortgages and they raise this money by issuing bonds or trading on the money markets.
So the mortgage referred to in the OP will be accounted for within a certain block of money in the lenders accounting system.
At the end of the 25 years the bank will have to pay back its investors. Fine, the banks have loads of money, but not if too many people don't stick to the agreement that they made to pay back all the money at the end of the term. That's how the bad loans in the US failed and why many Spanish banks have large liabilities that threaten their stability - they have lent on overpriced holiday apartments that have plummeted in value.
A spanner in the works for the 'victims' of the IO problems is that lending conditions have changed massively since the financial crash, so they can no longer roll these loans over indefinitely. They may be able to get a new loan, but they have to qualify in today's environment, which they don't always do if their income has dropped, or just tighter requirements.
In their favour, there are now probably more products available to older borrowers. After all, a pension income is far more secure than one from employment, and if the borrower dies, the property can be sold to repay the bank, so all is not lost.
However, it's risky relying on downsizing as a repayment strategy. It might work for some - those with larger properties in areas of high growth in value will probably be able to sell a large house, pay back the IO mortgage and be left with enough to buy a smaller property outright.
But it doesn't always work out like that. A year or two ago there was a similar story to the one referred in the OP where a woman had bought a new build flat in Liverpool at the previous peak, in around 1990 on an interest only mortgage, so it must have been in 2015 that she got to the 25 years and the bank wanted their money back. But by then, there had been the crash of the early 1990s and the 2007 crash and her flat, which had been an overpriced new build (she might have paid around £100k for it, which might not sound like much, but it does when you compare it with the 2 bed terrace that I bought in Leeds for £32k in 1995 - £100k for a small property in a northern city at that time was a lot), wasn't worth any more than it had been in 1990, so she had little or no equity.