The root cause here isn't really the Fed. We've been under-building housing since 1990, back when rates were 5-8%. If you track new housing starts against adult population, you'll see some pretty terrible things.
Granted this is lower than rates were in the 1980s. But if the answer here is you need to keep rates at 10-15% to build sufficient housing I don't see how that's politically tenable.
These are building starts (i.e. permits obtained, but construction not having broken ground yet), but the more complete US Census data at the .gov website shows that completions haven't been far behind starts within the last five years.
Low rates are effectively a demand-side subsidy for housing I think, similar to other policies. But as you point out, if you restrict the supply-side, demand-side subsidy can be counter-productive.
Granted this is lower than rates were in the 1980s. But if the answer here is you need to keep rates at 10-15% to build sufficient housing I don't see how that's politically tenable.