Could this tool be used to produce eg a statically-linked version of `curl` that would work the same on any Linux, without needing to build curl from sources?

I think you should look at https://github.com/intoli/exodus

> Painless relocation of Linux binaries–and all of their dependencies–without containers.

> Exodus handles bundling all of the binary's dependencies, compiling a statically linked wrapper for the executable that invokes the relocated linker directly.