While I believe strongly in the concept of Muslim unity and that we are meant to have brotherhood and sisterhood as one ummah, the reality is, there are rather significant differences in the Shi'i and Sunni heritages (theology, practice, hadith, scholarship, viewpoints on history, worldview, etc), as well as between different subgroups of Shi'is and different subgroups of Sunnis.
If one takes away the all differences, one is really left with very little that is substantial and/or a completely new ideology and new group. For instance, Shi'is and Sunnis have different collections of hadith, and will never entirely adopt each other's. If you take away all the hadith in the name of unity, then you are left with the Qur'an-only view of Islam, which is its own separate ideology.
Also, usually, "just Muslim" is just a code word for "Sunni". This is because what is done by the majority is seen as "normal" and what is done by the minority is seen as "weird" or "deviant". The vast majority of people who are "just Muslim" adopt an overall Sunni worldview and consider any Shi'i-specific views or practices - such as belief in the 12 Imams as an essential part of Islam or the Shi'i timings for breaking fast - to be deviant or optional.
So, there is no real merit to taking away the religious heritage in the name of unity. It is better just to encourage a spirit of tolerance and acceptance of diversity. That is, just because some of my beliefs and practices are different from those of a Sunnni does not mean that we cannot have mutual respect or friendship. In fact, true friendship is based on accepting differences, not forcing people to be identical.
I believe that if we have a spirit of tolerance, friendship, and acceptance of differences, then we can still have brotherhood and sisterhood in the ideal of the ummah. However, this spirit of tolerance needs to go both ways; it cannot be one-sided to work.
We all come together from different ethnicities and stand together to worship Allah together reciting the prayers in the same language. We stand beside eachother and move as one unit in prayer unified.