It may be just as you say. Or it may be that higher interest rates have a secondary result of lowering home prices. I'm of the belief that very low interest rates artificially inflate home prices. If rates are raised, then after a short time home prices might go down and make the house sellable (if the homeowner is willing to sell at a lower price, assuming of course his new home he's buying is also at a low price).
“very low interest rates artificially inflate home prices”
That is correct.
The past decade or so saw the rise of corporations using homes as investment vehicles—which did not increase the supply but did increase the demand.
That itself was due to low interest rates which made it cheap to raise capital for any venture—and investments were seeking anything that could get the a decent return.
Since most homes are not bought or sold in any given year a major increase in demand escalates the price in any given area in a hurry.
These days the quickest way to reduce home prices are a combination of massive construction and banning new purchases of homes by folks who do not live in them (or own only a few other homes they rent out—or some similar restriction).