BA hires pilots two ways, SSP and DEP. SSP is for low hour pilots straight out of flight school and they only hire these from integrated courses, no modular students at all. DEP is a program where BA hire pilots with a Type Rating and 400-500 hours away from other airlines. On DEP, they do take modular trained pilots.
You seriously misunderstand the nature of integrated courses. The reason they include fewer hours is because much of the hour building is done with an instructor aboard and the student logging PICUS. This allows an integrated student to complete the course in less hours (195 is the minimum, but I'm not sure if this includes the MCC). The modular student will get over 250 hours because they are required to have at least 100 hours of solo flight in order to apply for their CPL. Integrated training is denser and more intense. Both integrated and modular training routes lead to the exact ratings, but intergrated students get there in fewer hours because it is a more concentrated way to train. Virtually all airlines who have sponsored training in the past, those who currently have mentoring and tagging schemes with FTE, OAT and Cabair, and all major military flight schools, use integrated training. When airlines spend their own money on training, they seem to do it integrated.
Integrated training courses are more expensive than modular courses and some schools that run them put a lot of hype into marketing them and these are the primary reasons they get bad press, but don't confuse the bad press with the quality. I am willing to bet that many PPruners that writes negative posts about integrated courses and the companies that market them would eat their words and submit applications if airlines started paying for the training again. You can rest assured that if they ever do, they will choose integrated schools to provide it.
The survey is well known. It is the GAPAN/EPST survey and you can find it at www.gapan.org
. It is about 4 years old, so hardly what I would call current. They sent it to 10 airlines and three did not even bother sending it back. They do not reveal which airlines they sent it to. There are over 30 airlines in the UK, so 7 is hardly representative enough sample to conclude those over 30 should not bother.