Lived in South East (Deptford) for four years around 15 years ago. The area was (slowly) gentrifying but it was really interesting time to be there. Old classics (the market, pie and mash shop, "propa boozahs" etc) mixed with the new (restaurants, breweries etc).
Family situation meant we moved back up North but were it not for that, we'd probably still be there. Worked/stayed there quite a lot since, it's still a great city.
As someone who was (more) into going out at the time the options were endless, probably overwhelming to be honest. Frequently 6-7 bands on each night you'd want to see. I'm not sure that would still be my way of life there if I'm honest, and we'd probably have moved out of zone 2 by now (Walthamstow was always on the list).
In all my time of living there I rarely saw the attraction of "central" though, and can probably count on one hand the amount of times I ventured into town (not gig related). It really is a city of villages, and despite being 10 mins from Trafalgar Sq on the train we'd spend most of our time around Deptford, New Cross, Brockley and Greenwich.