Static linking doesn't actually solve any problem. Just use dynamic linking with (probably relative) rpath, and compile against a sufficiently old libc.
There's some common FUD about rpath being insecure, but that only applies if the binary is setuid (or otherwise privileged) and the rpath is writable by someone other than the binary's owner (all relative rpaths are writable since you can use symlinks; absolute rpaths are writable if they point to /tmp/ or a similar directory, which used to be common on buildbots).
This is really not hard; working around all static linking's quirks is harder.
There's some common FUD about rpath being insecure, but that only applies if the binary is setuid (or otherwise privileged) and the rpath is writable by someone other than the binary's owner (all relative rpaths are writable since you can use symlinks; absolute rpaths are writable if they point to /tmp/ or a similar directory, which used to be common on buildbots).
This is really not hard; working around all static linking's quirks is harder.