Why do you want to work at one of the big 4 tech companies? Software engineering is a very big field, you don't need to be really good at solving coding puzzles on a whiteboard to be a good developer. Hell, there are developers who do nothing but set up wordpress sites for their clients, and they're providing a needed, valuable service.
You can work as a web dev at a small company without being part of the mainstream silicon valley rat race. Sysadmin-type skills are pretty hard to hire for, and very useful at small companies. It sounds like you've got that going for you. I'd change your definition of success and focus on finding a job at a company with a culture you'd enjoy, and not worry so much about its perceived prestige.
You can work as a web dev at a small company without being part of the mainstream silicon valley rat race. Sysadmin-type skills are pretty hard to hire for, and very useful at small companies. It sounds like you've got that going for you. I'd change your definition of success and focus on finding a job at a company with a culture you'd enjoy, and not worry so much about its perceived prestige.