Is it valid for some mosques not to perform the first Adhan on Friday, based on the absence of the cause that necessitated it during the era of Uthman ibn Affan, may Allah be pleased with him?

The first adhan on Friday is a prescribed Sunnah for two reasons: First, it is from the Sunnah of the Rightly Guided Caliphs, and second, there is a consensus among the Companions and the Ummah regarding it. The claim that its reason has ceased is incorrect, because the reason still exists, which is to alert people to the approaching time of Jumu'ah prayer so they can prepare for it.