The problem is that command line architecture flags change the behavior of the compiler, but they don't change the precompiled runtime support code that is linked in to every executable.
So when you try to build armv6 binaries with a toolchain that is built for armv7, they don't work.
If you need to build static binaries without any dependencies, you can use the toolchains at https://musl.cc/.
I've managed to successfully build (on WSL, Ubuntu 18.04) with this toolchain: https://more.musl.cc/9.2.1/x86_64-linux-musl/
with settings -march=armv6zk -mcpu=arm1176jzf-s -mtune=arm1176jzf-s -mfpu=vfp