The well-known official position of the Shāfi`ī school of law is that `umrah is obligatory upon a Muslim, just like Hajj, once in a lifetime.
However, the majority of scholars hold the opinion that performing `umrah in one’s lifetime is not a religious obligation at all. This is the view of the Hanafī and Mālikī schools of thought. It is also one of the views expressed by Ahmad b. Hanbal. Al-Shāfi`ī also one time held this view, and it seems to have been his older opinion.
The stronger of the two opinions – and Allah knows best – is that `umrah is not an obligation. The Qur’an clearly mentions the obligation of the Hajj without once indicating anything similar for `umrah.
The Prophet further emphasized this by saying: “O people! Allah has enjoined the Hajj upon you, so go for Hajj.” [Sahīh Muslim (1337)]
He also did so by saying: “Islam is built upon five things: the testimony that there is no God but Allah and that Muhammad is His servant and Messenger, the establishment of prayer, the payment of Zakāh, the pilgrimage to the House, and the fast during the month of Ramadān.” [Sahīh al-Bukhārī (8) and Sahīh Muslim (16)]
As for specific hadīth on the topic of the ruling of `umrah, there are some which indicate that it is obligatory and others that indicate it is not obligatory. However, all of these hadīth have some weakness that can be levied against them.
The strongest evidence in support of `umrah being an obligation is a certain narration of the hadīth where `Ā’ishah asks: “Messenger of Allah! Do women have the obligation of jihād?”
The Prophet replied: “Yes, they do, but it involves no fighting: Hajj and `umrah.” [Sunan Ibn Mājah (2901)]
However, this wording is at variance to the well-known narrations of the hadīth which make no mention of `umrah, like the one found in Sahīh al-Bukhārī where `Ā’ishah says: “We see jihād as the best of deeds.” To which the Prophet replies: “But the best jihād is an accepted Hajj.” [Sahīh al-Bukhārī (1520)]
This puts into doubt the narration which mentions “`umrah”. It is most likely a mistaken narration. Indeed, all the hadīth discussing whether `umrah is or is not obligatory are weak hadīth.
Therefore, the question is best referred back to the default ruling on religious duties in the absence of specific evidence, which is that there is no obligation.
As for the argument given in your question that `umrah was combioned in the Hajj because it is also obligatory on its own, I would say that the rites of `umrah are all included within the rites of Hajj. This is the reason why the Prophet said: “`Umrah is incorporated in the Hajj.” [Sahīh Muslim (1218)]
Therefore, we say on the strength of the general evidence that `umrah is a Sunnah. It is not an obligation.
However, we should have no doubt that the rewards for performing `umrah are immense and that `umrah is a very important form of worship.
Abū Hurayrah relates that Allah’s Messenger said: “Performing two `umrahs is an expiation for the sins that occur between them, and the reward for an accepted Hajj is no less than Paradise.” [Sahīh al-Bukhārī (1773) and Sahīh Muslim (1349)]
And Allah knows best.