I think I'm qualified to comment here because I just bought a house in SE London. It's a 3 bed, completely developed and renovated ex-council house which cost 325k and it's in Zone 4, just 10 mins walk from a station. The transport links are good at commuter times (national rail to any of London Bridge, Waterloo, Charing Cross or Cannon Street) and the area is both residential and safe. We also have an absolutely amazing view of London from the lounge and master bedroom (we're on a hill). We are both higher tax payers and earn a combined income of 6 figures, but we struggled with the deposit and have just got a Help to Buy mortgage, which means that the repayments are sky high. Coupled with commuter costs, we're not rolling it in, but we havent had to give up saving or our standard lifestyle for this.
The only real downsides are that the trains can be fairly infrequent later in the evening (only 1 per half an hour) and there are no bars/restaurants/supermarkets (just corner shops) near our house. Having said that, we both work in the city, so why on earth do we need those things right next to our house when we walk past 100 of them every single day?
I've encountered a lot of snobbery at work from people I have told that I live in SE London. I can understand this in respect of places like Lewisham and Peckham, which have long-standing reputational issues, but I live in a very residential, quiet area which hardly anyone has even heard of because they dont deign to go SE of The Strand. When I tell people what our house cost, they invariably say "oh I'm so jealous", yet they then go and over-leverage themselves on 600k 1-bed flats in Highgate.
So it's complete rubbish that you HAVE to buy a 750k flat to live in London happily, but many people here will tell you that you do for no other reason than they are too snobbish or lazy to move just 2 zones out.