Adjustment of Home Values. If any given area is selling at levels higher than standard median income allows and that given area is experiencing a higher than average number of Foreclosures (or pick a reasonable threshold), requirements should be in place to ensure that if a home defaults, it doesn't go back to market at those same values, but lower. After all, if you go strictly by banks rules, housing costs shouldn't be over 35% of income to hopefully prevent this from reoccurring.

I'm not sure I understand. Firstly, banks can't offer their foreclosed assets as significantly less than market rate because homes are used as collateral in a mortgage, so they'd have to just stop offering mortgages, which means only cash buyers get homes

Secondly, even if a bank did offer the foreclosed homes much cheaper, it won't fix gentrification, it just means whenever a house gets foreclosed a bunch of cash buyers swoop in and flip a house back to market rate for a free profit (or realistically, hold a bidding war and end up paying close to market rate anyways)

It doesn't make logical sense suggesting a bank has any power to change market rate (whether that's an asset bubble or genuine demand)