You'll not be surprised to learn that the average response has been 'We do not comment of matters of recruiting policy'. In fact, most won't even give me that much - but they are reading this! I suspect that one or two who've specified 'integrated only' (or OAT, say, only) are basically ill-informed about the current training requirements and examination culture, and are too lazy to learn - at the moment, those few going that route can get what they want within those specifications. That won't last.....
As for High Wing Drifter's contention that 'ex-integrated [Chief Pilots] will prefer integrated [applicants]', that would be a hell of an achievement - the modular/integrated system has only been here a few years! For someone to have completed an integrated course, left, joined an airline, progressed to command, and made Chief Pilot in that time would be... highly dubious! But there's some truth in what he says, and I do believe that those Chief Pilots (or DFOs) who remember the CAP509 system (which in itself is probably after their time) might equate integrated courses with that. Even Chief Pilots can be pretty dumb, sometimes!
Scroggs