Is reminding people before the funeral prayer, while they are waiting, a reprehensible innovation (bid‘ah) if it is done to seize the opportunity of their gathering and to avoid idle talk?

There is no harm in delivering an admonition before the funeral prayer, because admonishing and reminding people is a religiously prescribed act and is not restricted to a specific time. The Prophet, peace and blessings be upon him, used to admonish his companions at suitable times when souls were receptive to admonition, such as the time of waiting for a funeral, during which souls contemplate the gravity of the situation. The Prophet, peace and blessings be upon him, admonished his companions at the grave, as narrated by Al-Bukhari. There is no harm in this, as long as it does not become a fixed custom that resembles a regular sermon.