I have to ask. Aside from the writers and Gaider and such. What stops my Inquisitor from booting Cullen to the curb? Not like my Advisers can overrule me can they?
It would take a good deal of effort to find someone else who has been as high-up in a relatively large army as Cullen has and who's between jobs the way Cullen was. And then there's the troop morale stuff that's already been mentioned. And for that matter my understanding is that until you get to Skyhold, Cullen's membership in the Inquisition is Leiliana and Cassandra's decision, not yours.
And while this gets into the writing and mechanics stuff you didn't want to get into, the scripting to fire him would be extra effort to write, and would very possibly require a new character to be written since the only other obvious candidate I can think of among the characters in previous games (Ser Cauthrien) is not only at least as easy to find issues with but also very possibly dead. And then there's writing all the new dialogue that would require.