Why are these results different about 'frequency shifting'?
These angular errors are in the order of $10^{-10}$. These are numerical inaccuracies caused by the fact that exp() can't be arbitrarily exact.

"Exact" would mean that exp(1i*2*pi*Fs*t) would just yield a vector [1 1 1 … 1]; if you'd hit that, the input to the fft would be identical, because multiplication with $1$ doesn't do anything.

So, always consider the error relative to the phenomenon you're observing. You wondered about something that was about 200 dB lower in amplitude than the observed quantity.