I did out the math in the Reddit comment, and you can run the numbers by Googling [mortgage calculator] and plugging them in. 12% is the equilibrium interest rate where the "home price = 3x income" and "housing expenses should be no more than 30% of gross" rules of thumb are both true. At 3% interest rates, a 3x home will cost you about 12% of monthly gross, and 30% of gross will get you 7.5x income. Those two rules of thumb cannot both be true at 3% interest rates.
With taxes and insurance, those numbers change to 3x = 17% of gross and 30% of gross = 5.4x, assuming California tax rates. Things like emergency funds, retirement savings, etc. are included in the "housing should cost no more than 30% of gross" rule of thumb.
Housing expenses should be no more than 30% gross isn't a rule of thumb, it's a maximum amount lenders will traditionally lend you. It's called the front end ratio.
Quoting investopedia "Lenders prefer the front-end ratio to be no more than 28% for most loans and no more than 31% for FHA loans."
Lenders don't care if you can retire or if your kids can go to college, they care if you are likely to default on your loan. So I wouldn't take the front end ratio as personal finance advice.
The 3x income housing advise in not related to the front end ratio. 3x income is the max you should borrow (according to the advise givers, you are of course free to disagree), the front end ratio is what you can borrow.
With taxes and insurance, those numbers change to 3x = 17% of gross and 30% of gross = 5.4x, assuming California tax rates. Things like emergency funds, retirement savings, etc. are included in the "housing should cost no more than 30% of gross" rule of thumb.