Rules of thumb like “mortgage amount should be less than 3x income” don’t make much sense because they ignore the interest rate.
They're fine as a rule of thumb. The individual should still run their own numbers to see if their limit is higher or lower.
Even at 2.5% interest, I have a hard time seeing anyone being able to responsibly buy a $450k house on $80k income. They might be able to swing it, but they'll be screwed as soon as a large unexpected expense comes up. Even a 2% property tax will eat close to 10% of their gross income. There's no way they could save for an emergency fund or properly fund a retirement account.
I assume that is simply the way 80%+ of people in the US expect to live since it has been their reality for decades.
At 9% interest, the monthly payment on a 240k home is $1,931. Mortgage interest rates were higher than this throughout the 1980’s. They got as high as 18% which would yield a $3,617 monthly payment.
This means that the person who followed the “3x your income” rule in the 1980’s has a harder time paying the bills than the person who bought the home for 450k with a low interest rate.
This is a badly-formed rule because it ignores a key variable - the interest rate. Sometimes, it will prevent someone from buying a home they can afford. Other times, it will cause them to buy a home they can’t afford. There’s no need for such a rule, because mortgage payment calculators that take into account the interest rate are easily available. It’s like picking a shoe size based on how much you weigh instead of measuring how big your feet are. If it gives you the right answer, that’s out of sheer luck.
Of course, it’s necessary to also take into account factors like taxes and other expenses which vary from place to place and person to person, but a simple mortgage payment calculator is a good place to start.