For me, this really underline how an already optimised website won't benefit much from it (especially the image recompression bit), and it may even be harmful.
On the other hand, it could work great by being properly configured for small websites that don't get much optimisations, and could really benefit from generic optimisation.
You still could benefit, because the optimizations are user agent-specific, and if you aggressively use caching with Nginx, then you don't need to make as many live calls to the backend to segment the cache per user agent type. PageSpeed is written in C and highly optimized so it has performance benefits in your case as well, I think. For example, there are user agents for which PageSpeed may decide to deliver your CSS as inline and small images - as data URIs. It also automatically (but optionally) does lazy loading of images, etc. It has beacons that are also implemented with it. It's perfect for CMS websites like it is in my usecase. It's not a panacea though.
For me, this really underline how an already optimised website won't benefit much from it (especially the image recompression bit), and it may even be harmful.
On the other hand, it could work great by being properly configured for small websites that don't get much optimisations, and could really benefit from generic optimisation.