Uh, confusion.  Indeed, "PATH= foo" finds foo in the current directory on bash.  I'm not sure how I ran the original example.  It seems that a default PATH is used, which includes at least "/bin" and ".".

The point I was making in the original post is still valid: "PATH= foo" appears to behave identically to "unset PATH && foo" in all cases I tried so far.  For example, for me both work with some local executable or with "ls" (which is in /bin), and neither works with "which" (which is in /usr/bin).