I used to donate to the LSE regularly, till I wrote saying that DS was really enjoying his time there, and they replied saying I was no longer eligible and took me off their mailing list. As if my small donations would have been able to buy any influence. One day I might even ask to get the begging letters again.
Part of the reason why I did is that they work quite hard to keep Graduates on board. When I was working overseas, being able to join local LSE graduates group was a good way of getting to know both locals and expats and generally less stuffy than the Oxford and Cambridge lot. And now I can I attend some interesting public lectures (back in the day when they were in person only I sometimes had access to priority booking), and the occasional social event (in the past they have run a good election night special), but also use of the library, and a central London alumni centre. I have yet to visit the latter, but may turn up to an upcoming open day which involves talks, afternoon tea and more. They also run various professional networking groups (lawyers, publishing etc) and a once in a decade or so, reunion weekend for graduates from my year group, which was surprisingly enjoyable and which people came a long way to attend.
DS, who followed me to LSE, has also benefitted. He is completing a PhD elsewhere, but as well as using the library seems to have been able to wangle some office space (possibly unofficially) when he is in London, which in turn has enabled him to keep up with his University contacts in an organic way, ie by bumping into them in corridors.
In short, I don't think it is about culture, or about nostalgia but more what you get out of retaining/renewing contact.