Success Rate of applying for a Msc in Econ and Policy at KCL?

I'm currently a third-year student at SOAS University of London, studying for a BA in Politics and Economics. I do not regret the subject area of my degree as I very much enjoy it, but I do regret choosing SOAS as literally almost every company/institution does not even know about SOAS' existence and it's very difficult finding any graduate jobs that does not immediately screen for top/target UNIs (I became severely depressed throughout my 2nd year and this summer after being constantly rejected for internships).
With that being said, I am still searching for graduate jobs, but I need to have a plan B in case everything doesn't work out, and I am unable to secure a graduate job/internship for next year. I was considering pursuing an MSc or an MA in a similar field and discovered KCL's MSc in Economics and Politics, which is a perfect fit for my profile.
Perhaps it's my low self-esteem these days, following a year of consistent rejections, but I have been reviewing the LinkedIn profiles of some current students enrolled in this program, and they all have had at least one relevant internship or work experience before being admitted into the programme. I have no work experience, despite a strong academic history. I have consistently achieved all As from GCSEs to A-Levels, and now I am averaging at least a 70% across all years and modules at university. I meet the education criteria, but I am concerned about being rejected due to my limited experience.
I also have ZERO experience within my UNI's societies, mainly because almost all of them have been dead silent for the past 2 years, until only recently, but I haven't been that active. I genuinely do not know how I can possibly stand out, and it's crushing my soul. If I'm going to pursue an MSc, I will need to attend a target or Russell Group university, regardless. What do I do? Is there any hope for me? And also, I don't think I have the mental and physical capacity to move outside of London to study because of my depression and fear for my mental health.