Outside of aviation, a lot of training courses in various industries follow the integrated model, when I did the training for my current career it was definitely "integrated", however I was an employee from day one, had a wage and pension etc. so didn't have to worry about funding the training or indeed my living costs.
In aviation, even the most generous training programmes don't pay much in the way of living costs during training unless you have very few commitments, additionally you don't get the protection and rights of employment from day one of training. That might be about to change with a new training programme mooted by one of the low-cost airlines.
Even with the sponsored programmes of the past couple of years , the majority of trainee pilots still have to pay for their own training which leads to a catch-22. If you don't have a wealthy family background then you need to get a job to save money for flight training, if you want to go down the integrated route it would need to be a very good job allowing you to save at least £10k per year if you wanted to get in before 30 unless investment returns were particularly good etc. Going down the modular route in this scenario would allow a trainee pilot to be qualified years earlier whilst staying in their job until receiving an offer from an airline. The integrated route would require someone to give up their lucrative career which has allowed them to save despite there being no guarantee of what the market would be like at the end of training. Getting back into the original career after an 18 month break might be a challenge too, especially given the gap would involve pilot training.
As a result I suspect self-funded integrated courses produce a very specific demographic of pilot, that is usually 18-25 from a wealthy family background.