Southwark is now full of purpose-built marginal tourist attractions. The big Ferris wheel. The springy Millenium Bridge. The Tate. The Globe Theater rebuild. And I see that the Oxo Tower has now put in a bar. Battersea itself now has a giant glass elevator that rises up one of the smokestacks and comes out the top. All the sights of historical importance are elsewhere.
I spent a few weeks in that area once. It's apparently less grungy now.
FWIW, most of the things you describe there are affordable and very grounded in London.
The Millennium Wheel, that is a tourist attraction, obviously, and is a slightly odd experience and not terribly cheap but it's accessible to everyone. The Globe Theatre is pretty grounded in London culture and is really affordable.
The springy bridge is a functional thing that goes between the City and the more affordable and touristy bits of Bankside.
Tate Modern is IMO a triumph. Free to enter and chock-full of properly amazing stuff that is free to view. (Go see "In Search of Vanished Blood" by Nalini Malani). There are other free galleries along that side including in the development around the Oxo tower.
For all the glitzy aspects and the touristy bits, it actually still feels at least a little like London (and is just round the corner from both normal working London and
heritage London).
None of that is even close to being as soul-deadened as the Battersea Power Station development, which exists outside culture.
I spent a few weeks in that area once. It's apparently less grungy now.