Aha, yeah, I've been found on a few
Love it! That's exactly the same attitude I take: not caring how competitive it is and just going for it. Sure, I might not get there but it's not the end of the world and there are other routes of getting to my desired destination.
Google, Facebook (I think they pay the same in London as they do in SV), Amazon, Microsoft (in Seattle), Uber, Snapchat, eBay, Adobe, Linkedin, Bloomberg, Dropbox, Palantir, Twitter and many more, all pay in that ballpark to software engineers, designers and 'product managers'.
Of course the public companies will have more 'liquidity' in terms of stock grants vs a private company, so your 'available total compensation' would be higher at a public company vs say, a smaller startup or private tech company that is yet to be acquired/IPO. That said, if the latter does IPO/get acquired, the upside is usually higher.