Dave
Unless you are working for an integrated school, as I am I admit for a modular school I cannot see why you have written what you have. I sat a modular course, having been offered and being able to afford integrated training. I cannot see any reason why someone would take integrated training, and pay more for less flying over a similar period of time to achieve the same (I also took my course full-time, and all delays would have equally affected integrated training). However I accept that some will.
I came out with the same licence as you will. Unless you get one of the very few jobs that are only available through direct contact with the schools (and many of us modulars have our contacts too) then you are better serve by the extra hours of a modular cchool than you are by the method of training, which potential employers willprobably not know. The only reason that integrated training is better for job prospects seems to be that the schools that offer both courses have been said to give preference to integrated graduates for those few recommendations they are asked to give - a recent and not entirely successful phenomenon. If they do then I suspect their motives; their modular students are not worse pilots unless the school has deliberately trained them less well, so why is that the case? Are they doing so just to sell the more expensive course?
In recent times a few airlines have had serious problems with students recommended by certain schools after training at various levels (fATPL (modular and integrated) or type rating). I anticipate that school recommendation will become even less important in airline recruitment, unless the school seriously improve procedures, including which course recommended students have completed.