We should take the advice given by the university Admissions very seriously. Universities like LSE explicitly says they evaluate based on PS, other than the normal grades etc, and they even specifically state what they are looking for, in other words, what they will be basing their evaluations on.
Sure, a good introduction helps but a well written PS is one that is able to communicate why and how YOU have decided and chosen this particular course. This may be less important in some universities but the competitive ones definitely will need the PS to help the Admissions Tutor select those who will complete the course and not back out after one term or after one year. That has implications for the student as well as financial implications to the university.
The PS would be one of the most challenging thing you would ever do. It challenges you to search and understand your own motivations to come up with something that is so PERSONAL that the Admissions Tutor would immediately know it is not another 'standard' PS.
Would the PS earn you a place if you lack some needed qualifications? For the competitive universities, it may be unlikely because firstly they have so many applicants and secondly the needed qualifications are there for a reason. Good luck.