Los Angeles once had a reputation for tearing down buildings without consideration of historical merit. No longer. Today only New York exceeds Los Angeles in the number of structures covered by historical designation. The trend extends throughout Los Angeles County, which contains 88 distinct local governments. That’s a victory for history, but not for homeowners, who say this patchwork of preservation programs has depressed housing prices and complicated sales.

Ellen Alperstein’s 1905 Santa Monica beach cottage has been...