NO - Obama did NOT “seek to unite the Muslim world.”
His engagement with the Muslim Brotherhood, particularly in Egypt, was not to “create peace in the mideast” or whatever BS slogan was used.
Obama’s support of the MB in Egypt, Hamas, etc... actually earned him the ire of much of “the Muslim World” particularly Saudi Arabia, Jordan, the Gulf States, and post MB Egypt. They hated him for it.
His main purpose was to raise up the Muslim Brotherhood’s various off-shoots and political lobbying organizations in the West (especially CAIR), to give them a higher political profile — to create another destructive, neo-marxist political wedge against the western institutions he hates.
CAIR are now the de-facto “voice of Islam” in the USA, thanks to Obama.
Muslim nations in the Middle East especially Egypt, Jordan and the Gulf States are rational. They are authoritarian states so they have a good handle of Islam in their countries.