What happens to software engineers who don’t climb the corporate ladder and stay as engineers?
They become better, more experienced, more sought-after, better-rewarded engineers. Some of the best big tech companies recognize this explicitly in their career ladders. At Google, for example, there is a separate job ladder for "individual contributors", which culminates (or once did) at "Google Fellow". An engineer at this level is commensurate with a Vice President, and has similar high-level impact — even moreso on questions of technology, architecture and technical direction. I believe there are similar positions at Facebook, Apple, etc.