The current locality rate for the DC-ish area (which actually goes halfway to Wilmington and Richmond) is 28.22%; thus, a GS-12-05 salary is $72k in a low-cost-of-living area and $92k in DC. The Federal Salary Council's recommended locality rate is 78.81%, bringing that notional GS-12 to $128k. So there is already somewhat of a disconnect between federal salaries and cost of living.

That aside, it's easy enough for the government to fix the housing problem. They're just too invested in keeping prices high to want that.