If you were right, then why wouldn't the data support it? Certainly, no one can doubt that the London School of Economics on average, gets great results: Economics is the highest-paid undergraduate degree, and the LSE gets a great lift there.
Unlike the LSE, Imperial has a separate business school with excellent resources including a dynamic alumni association and a large, long-established careers team. The LSE's department of management is far behind: it has a handful of people who manage alumni, careers skills and relationships with recruiters. Most of them were hired last year. It's coming very late. And, while terminology differs from college to college, it's just one department at the LSE rather than a whole faculty. The Bayes business school is one of six faculties at City. King's Business School is one of nine faculties; Imperial's business school is one of four faculties. They all have stunning, dedicated buildings. It's a core part of those colleges. However, at the LSE, Queen Mary and UCL the study of management is devoted to just one department: it has a lower level of resources. The department of management shares the Marshall building with LSE sports centre, cafe, philanthropy institute, music rooms and some other departments. It feels like the place at Waterloo that King's Business School was at 20 years ago, and City's business school was at 40 years ago in the Barbican. It has much less good facilities than ESCP London's beautiful campus in Hampstead.
For comparison, Imperial's Business School has 29 people working on careers and employer relations:
https://www.imperial.ac.uk/business-school/programmes/careers/our-team/ and seven people in its alumni team:
https://www.imperial.ac.uk/business-school/alumni/contact-us/ that's more than seven times more people than the LSE school of management, and that's before you compare the levels of skill of those people, which is also generally much lower at the LSE.
Momentum matters. Progression in the years right after your MSc are vital. I increased my salary 40% during my MSc, which I completed part-time at City, by getting an analyst role. In my first job change after graduating, I roughly doubled it, winning a role managing a small team of marketing specialists. Two or three years after that, I was working at Deloitte with a basic salary increase of around 40% and an on-target-earning package of more than four times my salary as an analyst. At that point in my career, which school I did my MSc at didn't matter as much. The LSE's fame doesn't matter much in the UK relative to how quickly you accelerate.
As you say, everyone is free to have an opinion. The LSE isn't a blot on your CV. Honestly, I would say that the LSE's management MSc students do very well relative to the modest support they get and the heavy research orientation of the department's scholars. However, you just don't have the same support in its department of management that you have at Imperial. Imperial's MiM has performed better well in the FT and QS rankings relative to the LSE, Warwick and Manchester. It will take a huge investment for the LSE to overcome that. The LSE doesn't intend that, looking at its 2030 strategic plan:
https://www.lse.ac.uk/2030/assets/pdf/LSE-2030full-text-as-approved-by-Council-5-Feb.pdf