You're a lot closer to everything in the Barbican than you are in Croydon or Enfield or Acton or Stratford.

London is big. The City is close enough to the centre that it is central, compared to most of London.

(Personally I think the Barbican is ugly, and I didn't like moving around in it, with long walkways forcing unnatural navigation. It only works, in so far as it works, due to a degree of elite mindshare capture keeping it owned and occupied by the wealthy. Put the same idea in Stratford and come back to somewhere far less pleasant in 20 years.)

Also I’d argue the Santander Cycles are a form of public transport (just not mass transit like buses or the tube)

https://tfl.gov.uk/modes/cycling/Santander-cycles

But honestly, you’re the first Londoner I’ve spoken to who considers 30 minutes by foot a long way away. Even by London standards, that’s close. For suburban dwellers, 30 minutes by foot wouldn’t even get them close to their nearest art gallery (and I don’t even mean “major” galleries either).

And your insistence on limiting things by “major” instances is odd. London has a strong culture of smaller independent amenities. Many of which are a lot closer than Soho and Southbank.

This is honestly the first time I’ve ever heard anyone complain about a zone one apartment being a long way from stuff.