we need to build a system that makes it more expensive to own homes if you already own a home,

I don’t know how most places work, but my state does this by assessing property at whatever rate, for everyone, but you get an exemption on part of the assessed value for your primary residence. My house is taxed as though it were worth $60k less because it’s where I live, not an investment or business . This seems like a good idea that would do exactly as you suggest (although that exemption should arguably be higher and should increase once in a while)