So "Does what school you train with matter"? I think that's been answered. In my experience - I would say when hiring times are "good" the bigger integrated schools MAY help to smooth that process and you MAY find a [better*?] job more quickly than elsewise. When times are depressed, as they have been for close on 3 years now, then no. It makes very little difference. You won't find it easy to find employment anywhere, regardless of who you trained with.
What I did want to pick up though is the point made by prophead. Be VERY wary when choosing your training method that almos everyone in aviation is intractable when it comes to choice of training. EVERYONE defends their own choice as if it's the ONLY correct choice to make.
The point Prophead ends on ("How many OAA grads have ended up paying Ryanair anyway") is very valid...but completely neglects to consider how many people got jobs full stop in that period, by whatever method. In the last three years (I know, I am one of them) fresh grads have had virtually no choice at all. When I failed the sim check at Ryanair, their split between mod/integrated students interviewed was something like 20/80. The successful percentage [at that time - believe it's subsequently go even tighter] was something like 40%. I'm NOT having a go at Prophead, but I do think that if you read that at face value you could be badly misled. You could easily assume that those students could just have easily gone modular and got the same end result, which is NOT true. Yes, the integrated students ended up going to Ryanair, an airline famous for taking mod students, but out of every hundred interviewees, only about 4 Modular were being hired, as opposed to 32 integrated. Not great odds on either route, not great to end up spending a further £20k plus on TR, but "the only game in town" - and where "getting a job" is the name of the game, then that is not to be underestimated...
* - 'Better' is a distinctly movable feast. What for one person may be a perfect job may well be a nightmare for someone else. In this context I refer to the possibility of going direct to a legacy/charter carrier as opposed to a smaller turboprop operator. I am NOT making any assertions that that's what the majority of people want to do, or that it's the right thing to do, or that...