Dubious or not, it was surprisingly hard. There were a few years where Cambridge economics was seemingly inaccessible for certain schools, then it suddenly became easier. Some very strong applicants were rejected. Warwick was also odd. The older brother of a friend of DDs was also rejected at the same time. Bursary pupil, first of his family to go to University, etc etc. Yet someone else with very strong languages but perhaps weaker maths, was offered a place. I guess each University knows what it wants.
These courses are very oversubscribed. Some get accepted, some rejected. The important lesson is that nothing can be assumed. And, given most don't interview, the PS is very important.
I think it is true to say that LSE like strong maths. DS worked out that he could have taken his economics degree with only two economics courses. (He did a couple more, but not many, and chose Masters options so he did not have to write a single essay.)
All illustration that economic courses can vary hugely. DH read PPE so was properly shocked at how mathematical DS' degree was. DH seems to think that economists should be literate. (I studied at LSE with double maths, so am happy to join DS in claiming that DH has it wrong!)
A large number of DS's LSE friends are now in the US. His network extends across most of the major economics departments. LSE was a very good starting point. For what he wanted it was the perfect place.