There are some Ahadith regarding this issue in Sunni books and some Shia hadith records and one may say that at first sight, Quranic verses have also addressed this subject, but we cannot come to a conclusion about this issue directly from the Quran. Also, there are other Ahadith that reject Eve being created from the ribs of Prophet Adam.
However, even if we accept the mentioned form of creation, it will be in no conflict with women’s potential for spiritual development and perfection.
There are Quranic verses that imply that the creation of the woman was after the creation of man and some imply that Allah created the woman from a part of man’s body.
The verses are as follows:
1- " یا أَیُّهَا النَّاسُ اتَّقُوا رَبَّکُمُ الَّذی خَلَقَکُمْ مِنْ نَفْسٍ واحِدَةٍ وَ خَلَقَ مِنْها زَوْجَها " 
2- " هُوَ الَّذی خَلَقَکُمْ مِنْ نَفْسٍ واحِدَةٍ وَ جَعَلَ مِنْها زَوْجَها لِیَسْکُنَ إِلَیْها " 
3- " خَلَقَکُمْ مِنْ نَفْسٍ واحِدَةٍ ثُمَّ جَعَلَ مِنْها زَوْجَها " 
One may conclude that Adam was created first and then his wife Eve was created from him judging from the apparent meaning of the verses, but as some of the Quranic Commentators have mentioned, these verses do not definitely say that Eve was created from the ribs of Adam (as it has been mentioned in some Ahadith). 
This is why we believe that one cannot conclude that the woman was created from the ribs of man. But contrary to what was mentioned in your question this matter does not clearly contradict the Quran's account.
At the same time, we must keep in mind that there are many Ahadith narrated in Sunni books, Sahih Bukhari being one of them, which mention the women being created from the ribs of man. These Ahadith are also narrated in some Shia books,  but in this regard two significant points need to be taken into consideration:
First: Even though there are a great number of these Ahadith, there are also Ahadith narrated in the authentic books of the Shia that sharply reject this account or suggest a different form of creation. Two of these Ahadith are as follows:
1- Zurarah Bin A'yan narrates that Imam Sadiq was asked about the creation of Eve and the Imam was told that some people believe Allah created Eve from the lowest bones of Adam's left rib!
The Imam responded: "Glory is to Allah above doing such an act! Does one who believes this theory assume that Allah was not able to create a spouse for Adam that was not from his ribs so that troublemakers would not be mistaken to think that Adam married a part of his own body? What an opinion this is that some people believe in?! May Allah judge between us and them Himself!  "
2- Amr Bin Abi Al-Mighdam narrates that his father said: "I asked Imam Baqir:" How did Allah create Eve?" The Imam responded: "What do people believe in this regard?" I replied: "They say that she was created from a rib of his ribs" He said: "They are lying! Was Allah unable to create her from something other than Adam's rib?" I responded: "May I be sacrificed for you! Than what was she created from?" He said: "My father has narrated from his fathers that they said:" The prophet said: “Allah had held a handful of clay in his hand and mixed it with his right hand, of course both of his hands are right  , then He created Adam from that clay and there was some extra clay left, then He created Eve from that extra clay." 
The great Allamah Majlisi says in regard to resolving the contradiction between these two groups of Ahadith: "The Ahadith that mention Eve being created from Adam's left ribs were either said out of Taqiyyah (dissimulation) and fear of opposing the accepted theory at that time or that (when they say that she was created from his ribs) they mean that she was created from the extra clay he was created from, not that she was really created from Adam." 
Second: We know that in a normal situation every person is born from a mother and father and his body originates from their bodies, but this does not create a spiritual advantage and privilege for the parents and there are many times that a child, that is virtually a part of their body, reaches higher standards in terms of spirituality and human values. Similar to what was mentioned, today the issue of cloning has been brought up in which it is possible to create a creature similar to its clone by using a part of its body, but even in this case the second creature will have a different level of spiritual value.
Knowing what was explained above, even if we accept that the woman was initially created from a part of the man’s body, or according to other theories that suggest the creation of women was of a lower level versus that of men, this will not limit them from reaching high obedience of Allah and spiritual standards like men or even be better than them.
1- The Physical Creation of Adam, Question 399 (website: 415).
2- The Quran and the Creation of Man, Question 5356 (website: 5578).
3- The Reason for Creating Adam Before Eve, Question 8488 (website: 8722).
 Tabatabai, Mohammad Hosein, Al-Mizan fi Tafsir Al-Quran, vol. 4, pg. 136.
 Majlisi, Muhammad Baqir, Biharul-Anwar, vol. 11, pg. 97, the chapter on Adam and Eve and how they were named and some of their characteristics and their creation etc., hadiths 2, 3, 5, 6, etc.
 Sheikh Saduq, Man la Yahduruhu Al-Faqih, vol. 3, pg. 379, hadith 4336.
 Clearly, this statement is metaphorical of Allah’s power, not that it is implying Allah’s materiality or possession of physical hands.
 Tafsir Ayashi, vol. 1, pg. 216.
 Majlisi, Muhammad Baqir, Biharul-Anwar, vol. 11, pg. 116.