Perhaps some dialogues are not well-suited to public debate. There are reasons why most of the significant philosophical and theological conversations happen in obscure journals, academic papers, and in the oral defenses of Ph.D. dissertations.
It is unrealistic to expect that two people debating an ancient question of ontology, sociology, philosophy, or theology could achieve synthesis, persuasion, or agreement in a few hours. Discovery in philosophical and theological arenas moves in tiny increments at a glacial pace. In debate, the topic must be limited to a very small portion of the entire field of knowledge. You can't quote all of the pertinent sources in the time allowed and judgments must necessarily be constrained.
Contemporary debaters often disagree on the topic of a particular debate. They may explicitly agree on the title; but, the nuances of the proposition allow for a wide range of interpretations. Consequently, it can seem like the two are arguing for and against two different questions. Each debater readily ignores the other's questions because, to them, it appears that the other has strayed off topic.
Debate, by its very structure is a competitive endeavor. Cooperative methods might better serve ultimate reality and would more quickly lead to incremental progression in agreement upon shared understanding. What if two people started at a place in which they were in agreement before looking for the differences?
Another regular problem in contemporary debate is the uneven scholasticism of those debating. The two may be literally mismatched in their abilities to study and debate or they may be equally shallow in their approach to the debate. In 2010, when Christopher Hitchens and Tony Blair debated, "Is Religion a Force for Good in the World?," we saw an author debate a former politician. Blair was educated at Oxford as a legal theorist while Hitchens earned an Oxford bachelor's degree in philosophy, politics and economics. Both were intelligent celebrities, but neither was a serious philosopher. The debate was popular and accessible but did not go deep into the academic arguments of the day.
Perhaps one of the most obvious weaknesses of debate formats is that they usually do not influence minds. Entrance polls versus exit polls of audiences at debates typically show little movement. Most people come away from a debate with the same opinion with which they entered.
Most of us have at one time or another been enamoured by the potential of a single debate to change the course of history; it is likely that we have been greatly disappointed by this possibility. What we learn from debates is that they seldom have a lasting impact. Arguments don't create followers; but, a great parable can lead someone to follow the master. This is one of the messages of the life of Jesus of Nazareth. He was a master of telling stories that changed hearts and minds. He modelled a life that impacted the world and caused others to imitate his ethics, morality, and spirituality.
Story-telling, poetry, art, and literature are better tools than debate. If you want to change hearts and minds, don't become a student of debate; instead, become a poet, a song-writer, a sculptor, an artist, a writer, or a film-maker. There is good evidence that these disciplines will have a greater and more lasting effect.