A direct drive turntable, in days of yore, was a very specialized
motor. Your stepper 7.5 degrees is probably so "notchy" that
you will have audible flutter ("wow" and "flutter" being specs
of interest to audiophiles, as much as raw frequency response
of the pickup).
A belt drive setup was always superior even after decades
of motor engineering, soaking up small torque ripples.
I would therefore look for controllers with micro-stepping
features. I'd look for an integrated motor control chip
that drives power FETs or bridge drivers. Although you
could probably do this in software on a uC platform (and
there may be hybrid products comprising a uC and a
stepper motor control engine) I'd expect a lot of learning
to be required (which may be the point of all this, but
learning as little as necessary expedites things). You
might even find an eval kit which matches your chosen
motor nicely, and relieves a lot of the hardware design
/ build / debug / cursing.