In some countries where the Athaan is banned over the public address system, how can Muslims derive the benefit of responding to the Muath-thin‟s Call

Answer

If the people are unable to hear the Athaan being recited at the Musjid, then they should recite the Athaan in their homes. They will then have the opportunity of also answering to the Athaan. When the time for Salaat sets in, one of the family members should recite the Athaan. The speaker system you mention while permissible is not the same as a person reciting the Athaan. The Athaan over the speaker is like a parrot calling the Athaan.