The empire was in a difficult position at the time and it couldn’t afford an entire line of bad rulers. Why weren’t they deposed by some noble coup or some sort of revolt if they were as underwhelming as history often portray them to be? Did the nobility tolerate them because they served their interest in some way? Were the byzantine nobles also pleased by being overrun by Turks and Slavs?
You do bring attention to a historiographic slant that has gone unchallenged for too long here so kudos for that. Isaakios II Angelos when he first came to the throne by leading the revolt against Andronikos Komnenos was initially a tyrant-killer and the liberator of Constantinople. To that end he was already a member of the Komnenoi family through his grandmother, Theodora the fourth daughter of Alexios Komnenos, and so given that the direct imperial line died with Manuel's son in 1183, Isaakios' claim was as good as any other.
Consider Isaakios' initial accession to the throne and he appears an actually reasonably successful and dynamic military man. Accounts of the usurpation involve him shooting with a bow the assassins sent by Andronikos like a Homeric hero from a rooftop and his success was a good sign that God was on your side, a highly important aspect of becoming Byzantine emperor. The patriarch sided with him and popular support rallied to him. So Isaakios actually wasn't in a terrible situation in 1185 and that probably answers some of your questions. There were a few Komnenoi scions and pretenders, (two pretending to be Alexios II) but these were shown to be fakes and Isaakios was an active military leader and this encouraged longevity. He campaigned into Bulgaria, the Morea and despite angering the West by allying with Saladin to preserve the peace, the Third Crusade passes through Byzantium without huge disruption.
Where it starts to go wrong with the Angeloi is that Isaakios' wars go wrong. A Bulgarian uprising of the magnitude of the Asens' hadn't occurred since Basil II's reign and post-Manzikert Byzantium just lacked the resources to contain it. The lack of resources began to pile up when Constantinople proved unable to protect the provinces and they began to look for local leaders rather than Byzantine government. Money wins back a fair amount of the nobility during Isaakios' reign but Alexios III's reign is almost unmitigated disaster from which most of the Angeloi's infamy comes. From 1195-1203 he proved unable to put down the revolts for which he had criticised his brother and also proved himself duplicitous for the blinding of his brother in 1195 when he promised him retirement, the strangulation of his nephew Alexios IV in 1204 and then the poisoning of his blinded brother Isaakios shortly after. These events entered the historical record and the Angeloi got this reputation for ineffectuality. The other side of the coin is the realpolitik situation. Bulgaria took away the Northern frontier of the empire and a trading dispute with Venice closed off a lot of funds. The situation was dire no matter who was on the throne and the Angeloi had the connections to claim the title, it was just mostly down to Alexios III and his ultimate clash with his brother that brought the Fourth Crusade to the walls from which the reputation comes.