Speaking as an airline employed modular route-ee, different things work for different people and all of them require some luck.
I personally did a PPL near my home, as well as a night rating (or qualification as it was) and IMC rating now IR(R), distance learned the ATPL exams and then went another European country to hour build and do the CPL and MEIR (based on price - although quality was considered as well after visiting a few schools).
I got a job flying piston twins and then moved on to an airline job in Europe.
I didn't go to university or do any higher level education past A levels. I occasionally feel like I should have but that is more a personal thing than because it has hindered me in any way in my career thus far (it hasn't).
I couldn't afford to go to CTC or Oxford or the likes, but I now work with some people who did as well as some people (like myself) who didn't. I know a good few guys that went the modular route and haven't found a job and likewise I know a good few guys who went through the CTC/Oxford route and haven't found a job.
My own opinion is that either route can lead to a job, but doing the integrated route required less research and effort to give the same chance of a job as a modular person. That said I think (it's a generalisation and there are of course exceptions) most modular students put a lot more effort in because they know they have to. I am still often surprised by the lack of industry knowledge and naivety of some of the guys and girls that come out of integrated schools who can't understand why the queue of airlines wanting to employ them hasn't appeared, and it usually takes them 6 months to a year of unemployment to buck up and do some work, by which stage the modular person could be 'ahead'. This is precisely how I got my first job after integrated friends of mine turned their noses up at it.
Going solo in a reasonable time was mentioned in this thread. I don't think that matters (apart from maybe in one of those mentored schemes were airlines watch progress). I have never been asked or had to tell anyone how many hours it took me to go solo.
Things that can (but not always) matter as far as getting a job go are passing skills tests first time and ATPL exam results (unfortunately - since they aren't always relevant).