Could you describe more fully the phase-shifted case? When I simulate
it, I get a nearly non-directional pattern... that's with two
dipoles, separated vertically by a tiny amount, with a current source
at the center of each. That's as I'd expect, since that configuration
has been used for a very long time as an omnidirectional horizontally
polarized antenna for things like FM broadcast reception.

But why not just vary the relative amplitudes in the two dipoles? Then
you can rotate the pattern fairly easily. (The pattern when both
elements are contributing won't be quite as sharp as a simple dipole,
but the null could be useful on receiving.) You can use a variometer
to change the relative amplitudes. A variometer consists of two coils
in mechanical quadrature, with the center point of the two coincident.
Then at the center, there's a rotating coil. So you feed power to the
rotating coil (or feed your receiver from it), and connect the two
fixed coils to the antennas.

It's a big help that the two dipoles in that orientation don't couple
to each other (at least in an "ideal" installation).

By the way, if you Google "variometer," you'll probably find references
to just one fixed coil with a rotatable one inside it. That's not the
only kind...maybe the one I described above has a name; I tried
"quadrature variometer" with no real luck.

Cheers,
Tom