Nuh uhhh.... unity in the hearts, then physical unity will come. The salaf as-salif did not need inventions and theorems to unite. What we are lacking is iman, and the initiative.
Totally with you on that. Complete agreement.
Notice the subtle difference between "unity so that we can be good people" and "being good people so that we can unify". And by "we" I mean "you" of course. Unification by any means first means oppression and war. You can hope that you all will be better Muslims at the end of it, but it is more likely that you will simply be oppressed. Also it is a "waiting for Godot" situation - if you have to wait for a Leader with Perfect Iman, you will be waiting a long time. It really means doing nothing. You have passed the buck to someone else. But being good Muslims first so that all other Muslims will want to join you without being forced to is different. For one thing it means you have to be good Muslims in order to attract other Muslims. You have to go on being good Muslims all the time or the other Muslims will desert you. Above all else it is something you can do now. You do not have to wait for the Mahdi or some lesser but still impressive figure. You can do it yourself. All you have to be is better Muslims and nicer people.