What is the ruling on the takbirat al-tabligh (the mu'adhdhin repeating the Imam's takbir during prayer) if the Imam's voice is audible via loudspeakers and the mu'adhdhin is performing takbir without them?

There is no need for a muballigh (one who relays the Imam's takbirs) if the Imam's voice reaches the worshippers via loudspeakers. Relaying the takbirs is prescribed if the Imam cannot make himself heard by the congregants due to a weak voice or a large number of worshippers. The evidence for this is that Abu Bakr used to make the Prophet's takbirs audible to the people when the Prophet fell ill.