This is a case of 'it depends '. The damage isn’t caused by something being on or off. It’s everywhere. Disconnecting can isolate damage from small storms, but world ending storms have enough energy to jump air gaps and the surges would be faster than most breakers can react to. You’d physically have to rip the cables from your house to be safe. Smaller, battery powered devices would be more susceptible regardless if they were on/off. Batteries are a concern because you don’t want them to incur damage/blow up due to an electric surge.

If this sounds overblown remember that during the Carrington Event, telegraph lines continued to operate for hours even after batteries were removed. In some cases, lines sparked and damaged equipment or personnel. These are very powerful storms that naturally induce electric current in circuitry.