That's not going to completely solve the issue you're describing, since not everyone receiving the stimulus funding actually needs it to pay for those cost of living expenses. For those who are out of work, sure: people with higher rent need more stimulus. But if I'm still working in Indiana and you're still working in Boston, and we're both making enough to cover our rent/mortgage, *then* why would you receiving more than me be an equitable solution?
For me it's because my good paying job in the higher rent area has me living paycheck to paycheck to just pay bills. I have to work side gigs on top my full time job to get by. Those side gigs don't exist right now. So my income is less. However, since I'm still working my full time job I don't quality for any unemployment and the stimulus is all I get. For many people living high cost of living areas, the side gig economy is crucial. It's how they get by.
I know the point you are trying to make. And it does make sense. But it's not always as simple as that.