So perhaps a sort of gentleman's club (not that sort) for the younger generation?

I'd love to have an accessible chill out place I could go to just to get some work done, preferably with a good view of something. No music playing, just quiet and the ability to get a coffee and simple food if necessary.

Google Campus in London does this, but isn't that comfortable for long periods (plus it's absolutely packed).

I use Regus business lounges for this. They have office spaces in most major cities all around the world (at least 15 in San Francisco alone). You pay a reasonable fee per month (around $100, although you can also get free if you look for the right offers) and you basically get access to a lite coworking space, with desks, power outlets, a kitchenette stocked with coffee/tea/snacks, good Wi-Fi, etc. Sometimes people use it for meetings (I've overheard more than a few VC pitches in San Francisco) or just getting things done. I also use it when I'm traveling in a place that doesn't have reliable Wi-Fi or clean water and I need a break to relax and be productive. Western style clean bathrooms in China are good perk as well!