I think that is actually a classic sunk cost argument.
The stamp duty, solicitors’ fees and money already spent on the house are gone. Staying for longer does not somehow recover them, and the market is under no obligation to repay those costs through future house price growth.
That is especially important in a weak and uncertain market. If prices stagnate or fall, they could sit there for years paying a much higher mortgage and still not get back what they have already spent on moving and improvements.
Selling also does not mean “throwing away the equity”. Apart from the costs of moving, the equity goes with them and can be used to reduce the size of the next mortgage.
And I still don’t think “tighten your belts and ride it out” is automatically the low risk option. If riding it out means no savings, no emergency buffer and relying on childcare falling, wages rising and mortgage rates improving at just the right time, that is a gamble too.
The sensible question is not “how much have we already spent on this house?” It is “from today onwards, which option leaves us in the strongest financial position?”
Sometimes the answer will be to stay. Sometimes it will be to downsize. But keeping an expensive house purely because you have already sunk a lot of money into it is exactly how people get trapped by sunk cost thinking.