I'm fairly certain you're confusing anecdotes and culture with "big tech vs. your local shop". Software being a people problem meaning big tech is more people focused is a false dichotomy.
Most of the big tech companies' engineers are doing the same type of work as outside big tech. However, their internal practices are scattered and faster than outside (IME), which means you have to be able to keep your head above water, drive your own career, and deliver impact. You need to understand the product space quite well, as you are expected to be a partner. Rather than delivering code, you are delivering value.
Maybe 1 in 20 engineers that I've worked with outside big tech had this trait. I think they set the tech bar high, but the real skill differentiation I have seen is ownership.
Software is a people problem and what big tech looks for is top notch technical skills and the mindset/ability to solve them in a chaotic environment.