Integrated courses are never appropriate to self-funded students.
The myths of greater chances and finishing more quickly are just that - myths. Even if the few airlines that preferred integrated (they still did accept modular) were still recruiting you have to take into account several negative factors.
Those operators took on a tiny number of inexperienced pilots. There are operators and flying opportunities for which integrated training is completely unsuitable, most GA, PPL instructing and all aerial work for example. I used to throw away every CV that said 'integrated' unless they had a lot of subsequent experience, as the training was just not suitable. These are area in which a lot of people built up hours to be attractive to bigger operators. Then you take into account the cost saving of modular training, which could be £40,000 depending on how you go about training. If you have this money it can be used to increase your chances with the airlines that do, historically, take on a lot of inexperienced pilots. Conversely if you are in unmanageable debt you just cannot join some airlines, as they require you to take out a loan to pay for a type rating. If your credit is bad you cannot take that job!
Listen to Auntie Whirls, she (almost) always talks sense.