I forgot how much of a nuisance Alexios III was, even after he lost the throne. The Angeloi were a mistake.
Kill all Angeloi!!!
But in all seriousness, I wouldn't exactly call the Palaiologoi a worthless dynasty, it ain't exactly a Empire saver, but it wasn't complete dog crap...well, at least not the early period, and is far better than the Angeloi by a country mile, but that's just semantics now...
In any case, Magnum is right. With a 1204 PoD, a united Byzantium can be forged in a myriad of ways, with a myriad of dynasties, be it the Angeloi, Palaiologoi, Kantakouzenoi, Laskari (from OTL), or those who never held the diadem, such as the Philanthropenoi or the Nemanjic (What would be a Hellenized form of that name?)
By the way, the Komnenos Doukas is just an Angeloi by any other name.