Did Omar forbid the narration of the Prophetic Hadith during his Caliphate, and did he prevent the Companions from doing so when he sent them to cities?

It has not been established that Umar ibn al-Khattab, may Allah be pleased with him, prohibited the narration of hadith. Rather, it is widely known (mutawatir) that he cared for the Sunnah of the Prophet, peace and blessings be upon him, both in learning and teaching. He narrated more than five hundred hadiths, and he would act upon the hadith he heard from the Companions. He also used to take turns with his neighbor in attending the Prophet's ﷺ gathering to hear hadiths. Furthermore, he encouraged caring for and learning authentic hadiths.

What has been established from him is the command to narrate few hadiths and not to narrate excessively, out of fear of error and delusion when narrating in abundance, and to give reverence to the hadith, and also so that people would not be preoccupied from the Book of Allah; for the Quran is the origin of all knowledge. Some scholars have interpreted this command to apply to hadiths that are not related to Sunnahs and obligatory duties.