Integrated courses show airlines you can absorb x amount of material in a defined period of time... If you complete the same course content as a modular student over 5 years, what have you really proven? Will you take 5 times longer to complete a type rating if they hire you?
If you can do the modular route full time over 12-18 months with just the one training provider, and save yourself 25%, then do that. And don't put on your cv whether you're modular/integrated...
As you've already mentioned, many airlines are using the likes of CTC for their low hour cadet requirements. You could write a list of airlines who take on independent unaffiliated cadets... I can't think of any off the top of my head.
Once past your first airline job, all of this becomes irrelevant. They just want to see a license and verified stamped logbook. But always stay current on type, whatever stage you are at. Enjoy the journey!