No, it’s not the right metric. Which is why people don’t use it.
Imagine you make $160K and buy the nicest house you can afford with that income.
Then you get married, and your spouse makes $100K. Your household income has increased to $260K, which means you can afford an even nicer house.
Your per capita income has decreased to $130K. By your logic, you can’t afford a nicer house. In fact, with a second income you might no longer be able to afford your current house. That’s nonsense.
When multiple people live in a house they all have the opportunity to contribute to paying for it. Some may contribute a lot, some (like children) may contribute nothing. The house you can afford depends on the total amount everyone contributes, aka household income.
if the cost of a house goes up
This doesn’t make sense. The cost of a house is fixed when you buy it. It won’t ever go up while you live there.