If the result of your sandboxed shell is that it can run any program in PATH, which is sort of the basic function of a shell, then what is the value of the sandbox? I'd say one core feature is the ability to use su or sudo to escalate privileges and install a new application, and then run that application. If you have some whitelist of PATH applications at _start, that's impossible. Meanwhile, you can run dd or cc or rm or chmod and do whatever you like. So I'm just not seeing the value of sandboxing a shell.
I think there's a lot of value in sandboxing more constrained applications, especially with untrusted inputs, like web browsers, individual commands, etc. But I don't buy that you can meaningfully reduce the privileges of a shell without losing the quintessential features of a shell.