Curiouslama, your sample may have been skewed by the fact you interned with students from the LSE, presumably in some sort of IB/IB adjacent role. Taking a similar sample of DS' friends the dominant ambition would have been some sort of research and forecasting role either in academia, or within a research consultancy/think tank.
The LSE recognised the interest in IB careers and a few years back started offering a dedicated Finance degree which presumably helps the the economics department focus on a broader study of economics.
In terms of social life, LSE did host one of the most popular student discos (Saucy) in town but looking for it, there seems to have been a problem with too many outsiders and some problem incidents, and the disco has moved to another location, though the bar remains. It is common for students to go to social events at other Universities. In my day the top places were the School of Mines and the London College of Printing, though I remember seeing a very stoned Nico at Imperial.
In some ways London has an advantage, students are scattered so social life tends to happen on campus which leads to societies being more active (our comparison is Bristol where only the would be politicians and actors ever seemed to go to the student union.) Sports fields are some way out, though the train journey from Waterloo is relatively straight forward. The advantage then is matches are mainly played against other London Universities. (DD at Bristol saw most of Wales and even got as far as Brighton for an away match.) Imperial is the richest University, probably followed by UCL and has the best facilities. LSE, UCL etc however have access to ULU which includes a good pool. (Note Imperial and UCL offer some generous bursaries - and if living in London is going to be a struggle it is worth potential students asking LSE if they can give them something.) The same applies to other student competitions, though DS' gaming teams were on to a hiding to nothing when pitched against Imperial or UCL.
One of DS' friends went to Imperial so in his first year DS joined one of the Imperial sports societies with him, though I don't know the mechanism. I do know that DD has turned out a couple of times for a non-competitive Imperial team a couple of times when no longer a student.
Another advantage is that the library, thus the campus is open 24 hours and remains busy though vacations. It is quite common for international students to join friends studying in London for vacations rather than finding themselves isolated in Bath or Cambridge or wherever. The pattern of the academic year is also good with two 12 week teaching terms and then a shorter exam term.