(your way is correct for 6.3% but roughly $30/month off for 6.75%)
If you have renters sharing your primary residence, you *may* have to pay income tax on whatever they’re paying you.
You’re losing some money because you have to make a down payment. You’ll probably have to put 20% down (or else you’ll be hit with higher interest rates). Lost interest income on 20% of a $500,000 house is around $250/month.
The whole thing is built for a single person ($5500 deduction, $7000 state tax) I’d argue that homebuyers are more commonly married than single.
